Simple Groups and the Classification Problem (Part

Simple groups and the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple Simple groups and the Classiﬁcation groups Simple-feasibility Using Sylow’s Problem (Part II) theorems for simple-infeasibility Properties of prime divisors Sylow-unique prime Vipul Naik divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime March 13, 2007 divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Outline the Classiﬁcation Problem (Part II) Basic notation and theorems Vipul Naik

Basic notation and Finding simple groups theorems

Simple-feasibility Finding simple Using Sylow’s theorems for simple-infeasibility groups Simple-feasibility Using Sylow’s theorems for Properties of prime divisors simple-infeasibility Sylow-unique prime divisors Properties of prime divisors Numbers with existence of unique Sylow subgroups Sylow-unique prime divisors Numbers with Core-nontrivial prime divisors existence of unique Sylow subgroups Closure-proper prime divisors Core-nontrivial prime divisors Closure-proper prime Forcing numbers divisors Concept of forcing Forcing numbers Concept of forcing Known results Known results Nilpotence

Nilpotence Meaning of solvability Meaning of solvability I Sylp(G) is the set of all subgroups of G whose order is pr (or equivalently, whose index is m). The elements of Sylp(G) are termed the p-Sylow subgroups of G.

I np = |Sylp(G)| is termed the Sylow number for p in G

Simple groups and Fixing some notation the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups r Simple-feasibility I G is a ﬁnite group of order N = p m where p is a Using Sylow’s theorems for prime, r ∈ N and m is an integer relatively prime to p simple-infeasibility Properties of prime divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability I np = |Sylp(G)| is termed the Sylow number for p in G

Simple groups and Fixing some notation the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups r Simple-feasibility I G is a ﬁnite group of order N = p m where p is a Using Sylow’s theorems for prime, r ∈ N and m is an integer relatively prime to p simple-infeasibility Properties of prime I Sylp(G) is the set of all subgroups of G whose order is divisors r Sylow-unique prime p (or equivalently, whose index is m). The elements of divisors Numbers with existence of unique Sylp(G) are termed the p-Sylow subgroups of G. Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Fixing some notation the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups r Simple-feasibility I G is a ﬁnite group of order N = p m where p is a Using Sylow’s theorems for prime, r ∈ N and m is an integer relatively prime to p simple-infeasibility Properties of prime I Sylp(G) is the set of all subgroups of G whose order is divisors r Sylow-unique prime p (or equivalently, whose index is m). The elements of divisors Numbers with existence of unique Sylp(G) are termed the p-Sylow subgroups of G. Sylow subgroups Core-nontrivial prime n = |Syl (G)| is termed the Sylow number for p in G divisors I p p Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability I np|m (the divisibility condition)

I np ≡ 1 mod p (the congruence condition)

I Every p-subgroup of G (subgroup whose order is a power of p) is contained in a p-Sylow subgroup

Simple groups and Sylow’s theorem the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Sylow’s theorems state the following: Simple-feasibility Using Sylow’s theorems for I Sylp(G) is a single conjugacy class of subgroups simple-infeasibility Properties of prime divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability I np ≡ 1 mod p (the congruence condition)

I Every p-subgroup of G (subgroup whose order is a power of p) is contained in a p-Sylow subgroup

Simple groups and Sylow’s theorem the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Sylow’s theorems state the following: Simple-feasibility Using Sylow’s theorems for I Sylp(G) is a single conjugacy class of subgroups simple-infeasibility Properties of prime I np|m (the divisibility condition) divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability I Every p-subgroup of G (subgroup whose order is a power of p) is contained in a p-Sylow subgroup

Simple groups and Sylow’s theorem the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Sylow’s theorems state the following: Simple-feasibility Using Sylow’s theorems for I Sylp(G) is a single conjugacy class of subgroups simple-infeasibility Properties of prime I np|m (the divisibility condition) divisors Sylow-unique prime n ≡ 1 mod p (the congruence condition) divisors I p Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Sylow’s theorem the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Sylow’s theorems state the following: Simple-feasibility Using Sylow’s theorems for I Sylp(G) is a single conjugacy class of subgroups simple-infeasibility Properties of prime I np|m (the divisibility condition) divisors Sylow-unique prime n ≡ 1 mod p (the congruence condition) divisors I p Numbers with existence of unique Every p-subgroup of G (subgroup whose order is a Sylow subgroups I Core-nontrivial prime divisors power of p) is contained in a p-Sylow subgroup Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Unique Sylow subgroup the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility Using Sylow’s theorems for If n = 1, then there is a unique p-Sylow subgroup. Since simple-infeasibility p Properties of prime the Sylow subgroups form a single conjugacy class, np = 1 divisors Sylow-unique prime forces the unique Sylow subgroup to be normal. divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Core for a prime divisor the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple The p-core of G, denoted as Op(G) is deﬁned in any of groups Simple-feasibility these equivalent ways: Using Sylow’s theorems for simple-infeasibility I It is the largest normal p-subgroup of G Properties of prime divisors It is the normal core of any p-Sylow subgroup Sylow-unique prime I divisors Numbers with It is the intersection of all p-Sylow subgroups existence of unique I Sylow subgroups Core-nontrivial prime divisors Thus, showing that the p-core is nontrivial is another route Closure-proper prime to ﬁnding a nontrivial normal subgroup. divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Outline the Classiﬁcation Problem (Part II) Basic notation and theorems Vipul Naik

Basic notation and Finding simple groups theorems

Nilpotence Meaning of solvability Meaning of solvability Call a natural number simple-feasible if the answer is yes, and simple-infeasible otherwise.

Simple groups and Simple-feasible number the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility Using Sylow’s theorems for The basic question we want to answer: Given a natural simple-infeasibility Properties of prime number N, is there a simple group of order N? divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Simple-feasible number the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility Using Sylow’s theorems for The basic question we want to answer: Given a natural simple-infeasibility Properties of prime number N, is there a simple group of order N? divisors Sylow-unique prime Call a natural number simple-feasible if the answer is yes, divisors Numbers with and simple-infeasible otherwise. existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Prime powers are simple-infeasible the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple We know that if p is a prime, then p is simple-feasible: the groups Simple-feasibility only group of order p is the cyclic group, and this is simple. Using Sylow’s theorems for It turns out for a group of order pr , the following are true: simple-infeasibility Properties of prime divisors I The center is nontrivial Sylow-unique prime divisors Numbers with I Every maximal subgroup has index p and is normal existence of unique Sylow subgroups Core-nontrivial prime I Every subgroup is subnormal divisors Closure-proper prime I Thus, the group clearly can’t be simple for r > 1 divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability The idea is to observe that the constraints on the nps in Sylow’s theorems are dependent only on N and not on any structural properties of the group. There are three ideas behind showing simple-infeasibility:

I Proving that there is some prime divisor p for which np = 1 from the congruence condition and the divisibility condition.

I Proving that some p-core is nontrivial

I Proving that the normal closure of some Sylow subgroup is proper

Simple groups and Using only the order the Classiﬁcation Problem (Part II)

Vipul Naik

We are given a number N with at least two prime factors Basic notation and and we want to use Sylow’s theorems to show that N is theorems Finding simple simple-infeasible. groups Simple-feasibility Using Sylow’s theorems for simple-infeasibility Properties of prime divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Using only the order the Classiﬁcation Problem (Part II)

Vipul Naik

We are given a number N with at least two prime factors Basic notation and and we want to use Sylow’s theorems to show that N is theorems Finding simple simple-infeasible. groups The idea is to observe that the constraints on the n s in Simple-feasibility p Using Sylow’s theorems for Sylow’s theorems are dependent only on N and not on any simple-infeasibility structural properties of the group. Properties of prime divisors There are three ideas behind showing simple-infeasibility: Sylow-unique prime divisors Numbers with existence of unique I Proving that there is some prime divisor p for which Sylow subgroups Core-nontrivial prime np = 1 from the congruence condition and the divisors Closure-proper prime divisibility condition. divisors Forcing numbers I Proving that some p-core is nontrivial Concept of forcing Known results I Proving that the normal closure of some Sylow Nilpotence

subgroup is proper Meaning of solvability Simple groups and Outline the Classiﬁcation Problem (Part II) Basic notation and theorems Vipul Naik

Basic notation and Finding simple groups theorems

Nilpotence Meaning of solvability Meaning of solvability One way of showing that N is not simple-feasible is to show the existence of Sylow-unique prime divisors.

Simple groups and Deﬁnition of Sylow-unique the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility Using Sylow’s Let N be a natural number. A prime divisor p of N is theorems for simple-infeasibility termed Sylow-unique(deﬁned) if for any group G of order N, Properties of prime there is a unique p-Sylow subgroup. divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Deﬁnition of Sylow-unique the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility Using Sylow’s Let N be a natural number. A prime divisor p of N is theorems for simple-infeasibility termed Sylow-unique(deﬁned) if for any group G of order N, Properties of prime there is a unique p-Sylow subgroup. divisors Sylow-unique prime divisors One way of showing that N is not simple-feasible is to show Numbers with existence of unique the existence of Sylow-unique prime divisors. Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability r I If N = p m and m does not have any prime factors that are congruent to 1 modulo p, then again p is again Sylow-unique. Thus, in a group of order pq, both p and q are Sylow-unique unless p|q − 1 or q|p − 1.

Simple groups and Some examples of Sylow-unique the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Meaning of solvability Simple groups and Some examples of Sylow-unique the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups r I If N = p m where p > m, then p is Sylow-unique. Simple-feasibility Using Sylow’s theorems for Thus, in particular, in N = pq for distinct primes p and simple-infeasibility q, then at least one of them is Sylow-unique. Properties of prime divisors r Sylow-unique prime I If N = p m and m does not have any prime factors that divisors Numbers with are congruent to 1 modulo p, then again p is again existence of unique Sylow subgroups Sylow-unique. Thus, in a group of order pq, both p and Core-nontrivial prime divisors Closure-proper prime q are Sylow-unique unless p|q − 1 or q|p − 1. divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability A somewhat weaker condition is the following: A number N is termed a USE-number or a Unique Sylow existence number if every group of order N has np = 1 for some prime divisor p of N (the prime divisor may vary with the group). Clearly, any USE-number is simple-infeasible.

Simple groups and USE-numbers the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Meaning of solvability Clearly, any USE-number is simple-infeasible.

Simple groups and USE-numbers the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple The existence of a Sylow-unique prime divisor is a very groups Simple-feasibility strong condition: it means that there is a single prime divisor Using Sylow’s theorems for p such that np = 1 for all groups of order N. simple-infeasibility Properties of prime A somewhat weaker condition is the following: divisors Sylow-unique prime A number N is termed a USE-number or a Unique Sylow divisors Numbers with existence number if every group of order N has np = 1 for existence of unique Sylow subgroups some prime divisor p of N (the prime divisor may vary with Core-nontrivial prime divisors Closure-proper prime the group). divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and USE-numbers the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple The existence of a Sylow-unique prime divisor is a very groups Simple-feasibility strong condition: it means that there is a single prime divisor Using Sylow’s theorems for p such that np = 1 for all groups of order N. simple-infeasibility Properties of prime A somewhat weaker condition is the following: divisors Sylow-unique prime A number N is termed a USE-number or a Unique Sylow divisors Numbers with existence number if every group of order N has np = 1 for existence of unique Sylow subgroups some prime divisor p of N (the prime divisor may vary with Core-nontrivial prime divisors Closure-proper prime the group). divisors Clearly, any USE-number is simple-infeasible. Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability However, this generalization does not help us directly yet. The reason is the congruence and divisibility conditions are independent for the prime divisors – there are no constraints relating the nps. So trying to show that the number is USE from the above methods will actually end up giving a Sylow-unique prime divisor.

Simple groups and The obvious way of showing USE the Classiﬁcation Problem (Part II)

Vipul Naik

Meaning of solvability Simple groups and The obvious way of showing USE the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems An obvious way of showing that a number is USE is to show Finding simple groups that all solutions to the congruence and divisibility Simple-feasibility Using Sylow’s conditions must satisfy n ≡ 1 mod p for some p (the p theorems for p simple-infeasibility may be diﬀerent for diﬀerent solutions). Properties of prime However, this generalization does not help us directly yet. divisors Sylow-unique prime divisors The reason is the congruence and divisibility conditions are Numbers with existence of unique independent for the prime divisors – there are no constraints Sylow subgroups Core-nontrivial prime divisors relating the nps. So trying to show that the number is USE Closure-proper prime from the above methods will actually end up giving a divisors Forcing numbers Sylow-unique prime divisor. Concept of forcing Known results Nilpotence

Meaning of solvability An example is 105.

Simple groups and A special case of square-free numbers the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

When N is a square-free number, we can use crude counting Finding simple groups techniques as well. The idea is that since all the Sylow Simple-feasibility Using Sylow’s subgroups of N are cyclic of prime order, they must have theorems for trivial pairwise intersection. Hence, we have: simple-infeasibility Properties of prime divisors X Sylow-unique prime N ≥ 1 + n (p − 1) divisors p Numbers with existence of unique p|N Sylow subgroups Core-nontrivial prime divisors This constraint, combined with the Sylow-uniqueness Closure-proper prime divisors constraint, shows that the number is USE. Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and A special case of square-free numbers the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Meaning of solvability The existence of core-nontrivial prime divisors again makes the number simple-infeasible.

Simple groups and Deﬁnition of core-nontrivial the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility Using Sylow’s theorems for A prime divisor p of N is termed core-nontrivial if the p-core simple-infeasibility Properties of prime of any group of order N is nontrivial. divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Deﬁnition of core-nontrivial the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility Using Sylow’s theorems for A prime divisor p of N is termed core-nontrivial if the p-core simple-infeasibility Properties of prime of any group of order N is nontrivial. divisors Sylow-unique prime The existence of core-nontrivial prime divisors again makes divisors Numbers with the number simple-infeasible. existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability We claim that unless p2|q − 1, p is core-nontrivial. There are three cases:

I np = 1. In that case, there is a unique Sylow subgroup, so it is itself the nontrivial core.

I There exist P, Q ∈ Sylp(G) such that P ∩ Q has order pr−1. In that case, P ∩ Q is normal in both P and Q, hence in G, so it is the nontrivial p-core r−2 I For every P, Q ∈ Sylp(G), |P ∩ Q| ≤ p . Then, we 2 2 get np ≡ 1 mod p and hence q ≡ 1 mod p

Simple groups and The normalizers technique the Classiﬁcation Problem (Part II)

Vipul Naik

r Basic notation and Suppose G is a group of order p q where p and q are theorems distinct primes and r > 1. Finding simple groups Simple-feasibility Using Sylow’s theorems for simple-infeasibility Properties of prime divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability There are three cases:

I np = 1. In that case, there is a unique Sylow subgroup, so it is itself the nontrivial core.

Simple groups and The normalizers technique the Classiﬁcation Problem (Part II)

Vipul Naik

r Basic notation and Suppose G is a group of order p q where p and q are theorems distinct primes and r > 1. Finding simple 2 groups We claim that unless p |q − 1, p is core-nontrivial. Simple-feasibility Using Sylow’s theorems for simple-infeasibility Properties of prime divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and The normalizers technique the Classiﬁcation Problem (Part II)

Vipul Naik

r Basic notation and Suppose G is a group of order p q where p and q are theorems distinct primes and r > 1. Finding simple 2 groups We claim that unless p |q − 1, p is core-nontrivial. Simple-feasibility Using Sylow’s There are three cases: theorems for simple-infeasibility Properties of prime I np = 1. In that case, there is a unique Sylow subgroup, divisors Sylow-unique prime so it is itself the nontrivial core. divisors Numbers with existence of unique I There exist P, Q ∈ Sylp(G) such that P ∩ Q has order Sylow subgroups r−1 Core-nontrivial prime p . In that case, P ∩ Q is normal in both P and Q, divisors Closure-proper prime hence in G, so it is the nontrivial p-core divisors r−2 Forcing numbers I For every P, Q ∈ Sylp(G), |P ∩ Q| ≤ p . Then, we Concept of forcing 2 2 Known results get np ≡ 1 mod p and hence q ≡ 1 mod p Nilpotence

Meaning of solvability Clearly, the existence of closure-proper prime divisors makes the number simple-infeasible. A somewhat weaker condition is to demand that the number is CPE or Closure-Proper Exists, viz for any group of that order, there exists a Sylow subgroup whose closure is proper (the prime may vary with the order of the group).

Simple groups and Deﬁnition of closure-proper the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple A prime divisor p of N is termed closure-proper if for any groups Simple-feasibility group G of order N, the closure of the p-Sylow subgroup is Using Sylow’s theorems for proper. simple-infeasibility Properties of prime divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability A somewhat weaker condition is to demand that the number is CPE or Closure-Proper Exists, viz for any group of that order, there exists a Sylow subgroup whose closure is proper (the prime may vary with the order of the group).

Simple groups and Deﬁnition of closure-proper the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple A prime divisor p of N is termed closure-proper if for any groups Simple-feasibility group G of order N, the closure of the p-Sylow subgroup is Using Sylow’s theorems for proper. simple-infeasibility Clearly, the existence of closure-proper prime divisors makes Properties of prime divisors Sylow-unique prime the number simple-infeasible. divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Deﬁnition of closure-proper the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple A prime divisor p of N is termed closure-proper if for any groups Simple-feasibility group G of order N, the closure of the p-Sylow subgroup is Using Sylow’s theorems for proper. simple-infeasibility Clearly, the existence of closure-proper prime divisors makes Properties of prime divisors Sylow-unique prime the number simple-infeasible. divisors Numbers with A somewhat weaker condition is to demand that the number existence of unique Sylow subgroups is CPE or Closure-Proper Exists, viz for any group of that Core-nontrivial prime divisors Closure-proper prime order, there exists a Sylow subgroup whose closure is proper divisors (the prime may vary with the order of the group). Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability We also know that the kernel is proper: it is in fact the normal core of H. The upshot is that we have shown that the Sylow-closure is proper.

Simple groups and Subgroups of small index the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems Suppose p|N and we have demonstrated the existence of a Finding simple groups subgroup H ≤ G of index m where m < p. Then, we have a Simple-feasibility Using Sylow’s natural map from G to Sym(G/H) = S . theorems for m simple-infeasibility Further, since the image of this map has order relatively Properties of prime prime to p, all elements in p-Sylow subgroups must map to divisors Sylow-unique prime divisors the identity. Hence, the kernel of this map contains the Numbers with existence of unique normal closure of any p-Sylow subgroup. Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Subgroups of small index the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems Suppose p|N and we have demonstrated the existence of a Finding simple groups subgroup H ≤ G of index m where m < p. Then, we have a Simple-feasibility Using Sylow’s natural map from G to Sym(G/H) = S . theorems for m simple-infeasibility Further, since the image of this map has order relatively Properties of prime prime to p, all elements in p-Sylow subgroups must map to divisors Sylow-unique prime divisors the identity. Hence, the kernel of this map contains the Numbers with existence of unique normal closure of any p-Sylow subgroup. Sylow subgroups Core-nontrivial prime divisors We also know that the kernel is proper: it is in fact the Closure-proper prime normal core of H. The upshot is that we have shown that divisors Forcing numbers the Sylow-closure is proper. Concept of forcing Known results Nilpotence

Meaning of solvability From the general theory of subgroups, np is equal to the index of NG (P) in G. Thus, if we can show that for any solution to the congruence and divisibility conditions, np < q for prime divisors p and q of N, we have shown that N is CFE, and hence, simple-infeasible. Conversely, to ensure that N is simple-feasible, we must impose the additional constraints on Sylow numbers that each of them is at least as big as the largest prime divisor.

Simple groups and Using Sylow numbers to ﬁnd them the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems Recall that np is the number of elements in the conjugacy Finding simple class of a given Sylow subgroup (say P). groups Simple-feasibility Using Sylow’s theorems for simple-infeasibility Properties of prime divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Thus, if we can show that for any solution to the congruence and divisibility conditions, np < q for prime divisors p and q of N, we have shown that N is CFE, and hence, simple-infeasible. Conversely, to ensure that N is simple-feasible, we must impose the additional constraints on Sylow numbers that each of them is at least as big as the largest prime divisor.

Simple groups and Using Sylow numbers to ﬁnd them the Classiﬁcation Problem (Part II)

Vipul Naik

Meaning of solvability Conversely, to ensure that N is simple-feasible, we must impose the additional constraints on Sylow numbers that each of them is at least as big as the largest prime divisor.

Simple groups and Using Sylow numbers to ﬁnd them the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems Recall that np is the number of elements in the conjugacy Finding simple class of a given Sylow subgroup (say P). groups Simple-feasibility From the general theory of subgroups, np is equal to the Using Sylow’s theorems for simple-infeasibility index of NG (P) in G. Properties of prime Thus, if we can show that for any solution to the congruence divisors Sylow-unique prime and divisibility conditions, np < q for prime divisors p and q divisors Numbers with existence of unique of N, we have shown that N is CFE, and hence, Sylow subgroups Core-nontrivial prime simple-infeasible. divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Using Sylow numbers to ﬁnd them the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems Recall that np is the number of elements in the conjugacy Finding simple class of a given Sylow subgroup (say P). groups Simple-feasibility From the general theory of subgroups, np is equal to the Using Sylow’s theorems for simple-infeasibility index of NG (P) in G. Properties of prime Thus, if we can show that for any solution to the congruence divisors Sylow-unique prime and divisibility conditions, np < q for prime divisors p and q divisors Numbers with existence of unique of N, we have shown that N is CFE, and hence, Sylow subgroups Core-nontrivial prime simple-infeasible. divisors Closure-proper prime Conversely, to ensure that N is simple-feasible, we must divisors impose the additional constraints on Sylow numbers that Forcing numbers Concept of forcing each of them is at least as big as the largest prime divisor. Known results Nilpotence

Meaning of solvability Simple groups and Outline the Classiﬁcation Problem (Part II) Basic notation and theorems Vipul Naik

Basic notation and Finding simple groups theorems

Nilpotence Meaning of solvability Meaning of solvability We thus talk of:

1. cyclicity-forcing number(deﬁned): Every group of that order must be cyclic (that is, there’s only one group upto isomorphism of that order).

2. Abelianness-forcing number(deﬁned): Every group of that order must be Abelian.

3. nilpotence-forcing number(deﬁned): Every group of that order must be nilpotent.

4. solvability-forcing number(deﬁned): Every group of that order must be solvable.

Simple groups and The order-forcing operator the Classiﬁcation Problem (Part II)

Meaning of solvability 2. Abelianness-forcing number(deﬁned): Every group of that order must be Abelian.

3. nilpotence-forcing number(deﬁned): Every group of that order must be nilpotent.

4. solvability-forcing number(deﬁned): Every group of that order must be solvable.

Simple groups and The order-forcing operator the Classiﬁcation Problem (Part II)

Meaning of solvability 3. nilpotence-forcing number(deﬁned): Every group of that order must be nilpotent.

4. solvability-forcing number(deﬁned): Every group of that order must be solvable.

Simple groups and The order-forcing operator the Classiﬁcation Problem (Part II)

Meaning of solvability 4. solvability-forcing number(deﬁned): Every group of that order must be solvable.

Simple groups and The order-forcing operator the Classiﬁcation Problem (Part II)

Vipul Naik Let a be a group property. Then the order-forcing operator Basic notation and on a gives a property of natural numbers, where N satisfies theorems Finding simple the property if every group of order N satisfies property a. groups We thus talk of: Simple-feasibility Using Sylow’s theorems for simple-infeasibility 1. cyclicity-forcing number(defined): Every group of that Properties of prime order must be cyclic (that is, there’s only one group divisors Sylow-unique prime upto isomorphism of that order). divisors Numbers with existence of unique 2. Abelianness-forcing number(defined): Every group of Sylow subgroups Core-nontrivial prime that order must be Abelian. divisors Closure-proper prime divisors (defined) 3. nilpotence-forcing number : Every group of that Forcing numbers order must be nilpotent. Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and The order-forcing operator the Classiﬁcation Problem (Part II)

I pq is cyclicity-forcing when p and q are primes with neither congruent to 1 modulo the other

I What happens if we take a product of three distinct primes?

Simple groups and For cyclicity-forcing the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and Every prime number is cyclicity-forcing, and any theorems Finding simple cyclicity-forcing number is square-free. groups Simple-feasibility Using Sylow’s theorems for simple-infeasibility Properties of prime divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and For cyclicity-forcing the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and Every prime number is cyclicity-forcing, and any theorems Finding simple cyclicity-forcing number is square-free. groups Simple-feasibility Given a square-free number, it is cyclicity-forcing if we can Using Sylow’s theorems for somehow show that for any group of that order, np = 1 for simple-infeasibility Properties of prime every prime divisor p. divisors Sylow-unique prime Thus, for instance: divisors Numbers with existence of unique pq is cyclicity-forcing when p and q are primes with Sylow subgroups I Core-nontrivial prime divisors neither congruent to 1 modulo the other Closure-proper prime divisors I What happens if we take a product of three distinct Forcing numbers primes? Concept of forcing Known results Nilpotence

Meaning of solvability Again, to prove that a number with every prime divisor having exponent at most 2 is Abelian, it is necessary and suﬃcient to show that each np = 1. Thus: 2 2 I Any group of order p q is Abelian unless p |q − 1 or q|p − 1.

Simple groups and For Abelianness-forcing the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Meaning of solvability 2 2 I Any group of order p q is Abelian unless p |q − 1 or q|p − 1.

Simple groups and For Abelianness-forcing the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple Every prime number and every square of a prime number of groups Simple-feasibility Abelianness-forcing. Further, for any Abelianness-forcing Using Sylow’s theorems for number, the exponent of any prime divisor is bounded by 2. simple-infeasibility Properties of prime Again, to prove that a number with every prime divisor divisors Sylow-unique prime having exponent at most 2 is Abelian, it is necessary and divisors Numbers with suﬃcient to show that each np = 1. Thus: existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and For Abelianness-forcing the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple Every prime number and every square of a prime number of groups Simple-feasibility Abelianness-forcing. Further, for any Abelianness-forcing Using Sylow’s theorems for number, the exponent of any prime divisor is bounded by 2. simple-infeasibility Properties of prime Again, to prove that a number with every prime divisor divisors Sylow-unique prime having exponent at most 2 is Abelian, it is necessary and divisors Numbers with suﬃcient to show that each np = 1. Thus: existence of unique Sylow subgroups 2 2 Core-nontrivial prime Any group of order p q is Abelian unless p |q − 1 or divisors I Closure-proper prime q|p − 1. divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Outline the Classiﬁcation Problem (Part II) Basic notation and theorems Vipul Naik

Basic notation and Finding simple groups theorems

Nilpotence Meaning of solvability Meaning of solvability Basic facts on nilpotent groups:

I Any subgroup of a nilpotent group is nilpotent

I Any quotient of a nilpotent group is nilpotent

I A direct product of nilpotent groups is nilpotent

Simple groups and Inductive deﬁnition of nilpotence the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple We deﬁne nilpotence inductively as follows: groups Simple-feasibility A ﬁnite group G is nilpotent if the center of G, viz Z(G) is Using Sylow’s theorems for nontrivial, and G/Z(G) is again nilpotent. simple-infeasibility Properties of prime divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Inductive deﬁnition of nilpotence the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple We deﬁne nilpotence inductively as follows: groups Simple-feasibility A ﬁnite group G is nilpotent if the center of G, viz Z(G) is Using Sylow’s theorems for nontrivial, and G/Z(G) is again nilpotent. simple-infeasibility Properties of prime Basic facts on nilpotent groups: divisors Sylow-unique prime divisors I Any subgroup of a nilpotent group is nilpotent Numbers with existence of unique Sylow subgroups I Any quotient of a nilpotent group is nilpotent Core-nontrivial prime divisors Closure-proper prime I A direct product of nilpotent groups is nilpotent divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability For this, let G be a group of order pr . Then the action of G on G deﬁned by g.h = ghg −1. The class equation for this action: X |G| = |Z(G)| + [G : CG (h)] where h vary over representatives of distinct conjugacy classes of size at least 1. Since each summand is divisible by p, so is |Z(G)|.

Simple groups and Groups of prime power order are nilpotent the Classiﬁcation Problem (Part II)

Vipul Naik

Meaning of solvability The class equation for this action: X |G| = |Z(G)| + [G : CG (h)] where h vary over representatives of distinct conjugacy classes of size at least 1. Since each summand is divisible by p, so is |Z(G)|.

Simple groups and Groups of prime power order are nilpotent the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and Since the property of having prime power order is closed theorems Finding simple under quotients, it suﬃces to show that the center of any groups Simple-feasibility group of prime power order is nontrivial. Using Sylow’s r theorems for For this, let G be a group of order p . Then the action of G simple-infeasibility on G deﬁned by g.h = ghg −1. Properties of prime divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Since each summand is divisible by p, so is |Z(G)|.

Simple groups and Groups of prime power order are nilpotent the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and Since the property of having prime power order is closed theorems Finding simple under quotients, it suﬃces to show that the center of any groups Simple-feasibility group of prime power order is nontrivial. Using Sylow’s r theorems for For this, let G be a group of order p . Then the action of G simple-infeasibility on G deﬁned by g.h = ghg −1. Properties of prime divisors The class equation for this action: Sylow-unique prime divisors Numbers with existence of unique X Sylow subgroups |G| = |Z(G)| + [G : CG (h)] Core-nontrivial prime divisors Closure-proper prime where h vary over representatives of distinct conjugacy divisors Forcing numbers classes of size at least 1. Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Groups of prime power order are nilpotent the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and Since the property of having prime power order is closed theorems Finding simple under quotients, it suﬃces to show that the center of any groups Simple-feasibility group of prime power order is nontrivial. Using Sylow’s r theorems for For this, let G be a group of order p . Then the action of G simple-infeasibility on G deﬁned by g.h = ghg −1. Properties of prime divisors The class equation for this action: Sylow-unique prime divisors Numbers with existence of unique X Sylow subgroups |G| = |Z(G)| + [G : CG (h)] Core-nontrivial prime divisors Closure-proper prime where h vary over representatives of distinct conjugacy divisors Forcing numbers classes of size at least 1. Concept of forcing Since each summand is divisible by p, so is |Z(G)|. Known results Nilpotence

Meaning of solvability Simple groups and Nilpotence for ﬁnite groups the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Clearly any group that is a direct product of its Sylow Simple-feasibility Using Sylow’s subgroups is nilpotent. It turns out that the converse is also theorems for simple-infeasibility true. Viz, a group is nilpotent if and only if: Properties of prime divisors Sylow-unique prime I It is a direct product of its Sylow subgroups divisors Numbers with existence of unique I Every Sylow subgroup is normal Sylow subgroups Core-nontrivial prime n = 1 for every prime p dividing the order of the group divisors I p Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Equivalently, a number is nilpotence-forcing if and only if:

I Every prime divisor is Sylow-unique

I Every prime divisor is Sylow-direct

Simple groups and Nilpotence-forcing number the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility A number is nilpotence-forcing if every group whose order is Using Sylow’s theorems for that number, is nilpotent. simple-infeasibility Properties of prime divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability I Every prime divisor is Sylow-unique

I Every prime divisor is Sylow-direct

Simple groups and Nilpotence-forcing number the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility A number is nilpotence-forcing if every group whose order is Using Sylow’s theorems for that number, is nilpotent. simple-infeasibility Properties of prime Equivalently, a number is nilpotence-forcing if and only if: divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability I Every prime divisor is Sylow-direct

Simple groups and Nilpotence-forcing number the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility A number is nilpotence-forcing if every group whose order is Using Sylow’s theorems for that number, is nilpotent. simple-infeasibility Properties of prime Equivalently, a number is nilpotence-forcing if and only if: divisors Sylow-unique prime divisors I Every prime divisor is Sylow-unique Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Nilpotence-forcing number the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility A number is nilpotence-forcing if every group whose order is Using Sylow’s theorems for that number, is nilpotent. simple-infeasibility Properties of prime Equivalently, a number is nilpotence-forcing if and only if: divisors Sylow-unique prime divisors I Every prime divisor is Sylow-unique Numbers with existence of unique Sylow subgroups I Every prime divisor is Sylow-direct Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Nilpotence-forcing number the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility A number is nilpotence-forcing if every group whose order is Using Sylow’s theorems for that number, is nilpotent. simple-infeasibility Properties of prime Equivalently, a number is nilpotence-forcing if and only if: divisors Sylow-unique prime divisors I Every prime divisor is Sylow-unique Numbers with existence of unique Sylow subgroups I Every prime divisor is Sylow-direct Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Outline the Classiﬁcation Problem (Part II) Basic notation and theorems Vipul Naik

Basic notation and Finding simple groups theorems

Nilpotence Meaning of solvability Meaning of solvability The only possibility for a solvable simple group is a cyclic group of prime order.

Simple groups and Deﬁnition of solvability the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups A group G is said to be solvable(deﬁned) if there is a sequence Simple-feasibility Using Sylow’s of subgroups: theorems for simple-infeasibility Properties of prime 1 = H H ... H = G divisors 0 E 1 E E n Sylow-unique prime divisors Numbers with such that each Hi+1/Hi is Abelian. existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Deﬁnition of solvability the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups A group G is said to be solvable(deﬁned) if there is a sequence Simple-feasibility Using Sylow’s of subgroups: theorems for simple-infeasibility Properties of prime 1 = H H ... H = G divisors 0 E 1 E E n Sylow-unique prime divisors Numbers with such that each Hi+1/Hi is Abelian. existence of unique Sylow subgroups Core-nontrivial prime The only possibility for a solvable simple group is a cyclic divisors Closure-proper prime group of prime order. divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability The holomorph of any cyclic group is an example of a solvable group which is not nilpotent.

Simple groups and Checking for solvability the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility Using Sylow’s Unlike the conditions of being cyclic, Abelian and nilpotent, theorems for simple-infeasibility solvability cannot simply be checked by looking at the Sylow Properties of prime numbers. divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Checking for solvability the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility Using Sylow’s Unlike the conditions of being cyclic, Abelian and nilpotent, theorems for simple-infeasibility solvability cannot simply be checked by looking at the Sylow Properties of prime numbers. divisors Sylow-unique prime divisors The holomorph of any cyclic group is an example of a Numbers with existence of unique solvable group which is not nilpotent. Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Solvability-forcing means simple-infeasible the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility Using Sylow’s theorems for Suppose N is a natural number which is not prime. Then, if simple-infeasibility Properties of prime N is solvability-forcing, every group of order N is solvable. divisors Sylow-unique prime Thus, in particular, there cannot be any simple group of divisors Numbers with order N, and hence N is simple-infeasible. existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability However, if we are told that every non-prime factor of N is also simple-infeasible, then we can conclude that N is solvability-forcing. We thus have the following: N is solvability-forcing ⇐⇒ every non-prime factor of N is simple-infeasible.

Simple groups and A partial converse the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple Suppose N is a simple-infeasible number. Can we conclude groups Simple-feasibility Using Sylow’s that N is solvability-forcing? Not necessarily. theorems for simple-infeasibility Properties of prime divisors Sylow-unique prime divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability We thus have the following: N is solvability-forcing ⇐⇒ every non-prime factor of N is simple-infeasible.

Simple groups and A partial converse the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple Suppose N is a simple-infeasible number. Can we conclude groups Simple-feasibility Using Sylow’s that N is solvability-forcing? Not necessarily. theorems for However, if we are told that every non-prime factor of N is simple-infeasibility Properties of prime also simple-infeasible, then we can conclude that N is divisors Sylow-unique prime solvability-forcing. divisors Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and A partial converse the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple Suppose N is a simple-infeasible number. Can we conclude groups Simple-feasibility Using Sylow’s that N is solvability-forcing? Not necessarily. theorems for However, if we are told that every non-prime factor of N is simple-infeasibility Properties of prime also simple-infeasible, then we can conclude that N is divisors Sylow-unique prime solvability-forcing. divisors Numbers with existence of unique We thus have the following: Sylow subgroups Core-nontrivial prime N is solvability-forcing ⇐⇒ every non-prime factor of N is divisors Closure-proper prime simple-infeasible. divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and Burnside’s theorem and solvability-forcing the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups Simple-feasibility We mention Burnside’s theorem: Any number with only two Using Sylow’s theorems for prime factors is solvability-forcing. simple-infeasibility Properties of prime Note that since the property of having at most two prime divisors Sylow-unique prime factors is inherited by prime divisors, it suﬃces to prove that divisors Numbers with existence of unique any non-prime number which has at most two prime divisors, Sylow subgroups Core-nontrivial prime is simple-infeasible. divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Now let G be a group of order paqb. Let P be a p-Sylow subgroup. Let g ∈ Z(P) be a nontrivial element. Then, CG (g) ≥ P. Thus [G : CG (g)] is a power of q, hence the conjugacy class of g in P. Thus, G cannot be simple.

Simple groups and How we prove this the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Meaning of solvability Simple groups and How we prove this the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple groups We use the following result from representation theory: Simple-feasibility Using Sylow’s theorems for If a group has a conjugacy class whose size is a nontrivial simple-infeasibility power of a prime, then the group is not simple. Properties of prime a b divisors Now let G be a group of order p q . Let P be a p-Sylow Sylow-unique prime divisors subgroup. Let g ∈ Z(P) be a nontrivial element. Then, Numbers with existence of unique Sylow subgroups CG (g) ≥ P. Thus [G : CG (g)] is a power of q, hence the Core-nontrivial prime divisors conjugacy class of g in P. Thus, G cannot be simple. Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability I Equivalently, there is no simple group of odd order (that is, odd implies simple-infeasible)

Simple groups and The Feit-Thompson theorem the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple Starting from a rather weak result like Burnside’s theorem, groups Simple-feasibility Using Sylow’s and with many many intermediate steps, Feit and Thompson theorems for proved the following astonishing result: simple-infeasibility Properties of prime divisors I Every group of odd order is solvable (that is, odd Sylow-unique prime divisors implies solvability-forcing) Numbers with existence of unique Sylow subgroups Core-nontrivial prime divisors Closure-proper prime divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability Simple groups and The Feit-Thompson theorem the Classiﬁcation Problem (Part II)

Vipul Naik

Basic notation and theorems

Finding simple Starting from a rather weak result like Burnside’s theorem, groups Simple-feasibility Using Sylow’s and with many many intermediate steps, Feit and Thompson theorems for proved the following astonishing result: simple-infeasibility Properties of prime divisors I Every group of odd order is solvable (that is, odd Sylow-unique prime divisors implies solvability-forcing) Numbers with existence of unique Sylow subgroups Equivalently, there is no simple group of odd order (that Core-nontrivial prime I divisors Closure-proper prime is, odd implies simple-infeasible) divisors Forcing numbers Concept of forcing Known results Nilpotence

Meaning of solvability