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An Introduction to Finite Simple Groups

Bryan Crompton Abstract Algebra Fall 2010 University of Massachussets Lowell Recall Some Definitions A set G with a binary operation is a if (1) associativity holds, (2) the identity property holds, (3) and the inverse property holds.

A H of a group G is a subset of G and also a group.

A H of a group G is a subgroup such that gHg −1 = H for all g ∈ H.

A group is called simple if its only normal are {1} and itself.

This doesn’t seem like a lot to ask, but trying to understand what all simple groups there are leads to fascinating complexities! Who cares about finite simple groups?

I do! But...

•There are only two possible homomorphisms from a . Since ker f is normal in G, and it can only be {1} or G, any homomorphic is G or {1} by the First Isomorphic Theorem. So these groups are simple.

• They are like primes and irreducible polynomials, but instead are fundamental building blocks for groups.

• Given all finite simple groups, every other finite group can be generated.

• There is a fascinating classification theorem! Jordan-Holder Theorem for Finite Groups

Let G be a group and consider a sequence of normal groups

G = G0 B G1 B G2 B ··· B Gr−1 B Gr = {1} called a composition series and where each Gi+1/Gi , called a composition factor, is simple. Then the theorem states that given a group, there is a unique (unordered) set of composition factors.

Compare with 156 B 52 B 4 B 2 B 1 where ratios of succesive numbers are prime (3, 13, 2, 2).

However, two groups can have the same composition factors! Going the other way (Holder’s Extension Problem)

Holder’s Extension problem is the following. Given two groups K and Q, how many groups G are there such that K C G and G/K ∼= Q?

There may be more than one! Example: Let K = Z3 and ∼ ∼ Q = Z2. Then S3/Z3 = Z2 and Z6/Z3 = Z2. Both S3 and Z6 work.

This is where the analogy with the primes breaks down.

There is a procedure to find all G in general. The Classification Theorem

The classification theorem, also called the enormous theorem, states that all finite simple groups belong to one of the following groups:

(1) A prime group, (2) An of degree five or greater, (3) A Lie type group, or (4) One of the sporadic groups. Prime Order Groups

Theorem. Let G be a finite group. Then the following statements are equivalent. (1) G has a prime order. (2) G is cyclic and simple. (3) G is abelian and simple.

For (1): Given a group G with prime order p, by Lagrange’s theorem, the only subgroups are {1} and G. So it has to be simple!

For (2): Recall that all prime order groups are cyclic.

For (3): Use Cauchy’s Theorem. That is, there exists a subgroup of order p for each prime p dividing the order n of G. Alternating Groups

Recall the Sn. The alternating group An is defined as the set of all permutations made up of an even number of transpositions.

Theorem. For n ≥ 5, An is simple. Outline of Proof. Lemma # 1. The set of all three cycles generate An. Lemma # 2. If a normal subgroup N C An contains at least one three cycle, it contains all three cycles.

We assume we have a nontrivial normal subgroup N of An, and then proceed to show it is An. This is done by showing it contains a three cycle using normality by considering several different cases. Some Additional Comments

Prime order and alternating simple groups are the easy ones! After these, we start to get farther afield.

Proofs of these facts are found in Rotman, Chapter 2, Section 2.8 entitled ”Group Actions.” A slightly different approach is taken there.

In Rotman, it is mentioned that the simplicity of A5 is part of the funamental reason why the quintic is unsolvable by radicals. Classical Groups (Lie type groups)

There are five classical groups:

(1) The (2) The (3) The (4) The sympletic group (5) The

If you recognize some of the terms, you shouldn’t be surprised that these terms come from linear algebra!

These groups aren’t themselves simple. The General Linear Group I believe the general linear group was briefly mentioned in class–it is also in Rotman.

Given a field K, the general linear group is

GLn(K) = {all n × n matrices with elements in K}.

There is a nicer way to talk about this using more explicit linear algebra terms. Recall a linear transformation from a vector space to itself is a map

T : V → V

that preserves the linear structure

T (ax + by) = aT (x) + bT (y). The General Linear Group (continued)

An automorphism on a vector space V is an invertible (or bijective) linear transformation from V to V .

The general linear group can also be described as the set of all linear automorphisms on a vector space V .

(Recall the set of all group automorphisms form a group as well–homework 2.92.)

If we have a finite basis v1, v2,..., vn for V , elements of V can be expressed as n-tuples in K n which allows linear transformations to be expressed as n × n matrices. Special Linear Group

The special linear group, SLn(K), is the set of all elements in the general linear group such that the determinant of the corresponding matrix is 1. Forms of Vectors Spaces

For a vector space V over a field K, a form on V is a mapping f : V × V → K such that

f (x1 + x2, y) = f (x1, y) + f (x2, y)

and f (x, y1 + y2) = f (x, y1) + f (x, y2). An inner product is an example of a form.

The automorphism group of a form f is the set of all linear transformations T : V → V such that f (Tx, Ty) = f (x, y). The Other Classical Groups

A symmetric bilinear form satisfies f (y, x) = f (x, y) and f (λx, y) = λf (x, y) for all λ ∈ K. We get the orthogonal group On(K, f ) from this.

A skew-symmetric bilinear form satisfies f (x, y) = −f (y, x) and f (λx, y) = λf (x, y) for all λ ∈ K. We get the sympletic group Spn(K, f ) from this.

If K is a field where with a certain “conjugation” operation, then a form Hermitian (sesquilinear) if f (y, x) = f (x, y) and f (λx, y) = λf (x, y) for all λ ∈ K. We get the unitary group Un(K, f ) from this. The Main Structure Theorem (obtaining simple groups) Given a group G, define the set S = {xyx −1y −1|x, y ∈ G}. Then the commutator subgroup G 0 is defined as the subgroup generated by this set or

G 0 = hSi.

The center of a group is defined as

Z(G) = {z ∈ G|zg = gz for all g ∈ G}.

The Main Structure Theorem states that given a classical finite group G, then if G 0 is the commutator group of G, in most cases G 0/Z(G 0) is simple. We must have a underlying finite field K, of course. Some Final Comments on Lie types

In total, the main structure theorem gives us six infinite families of finite simple groups. They include

PSLn(K) = SLn(K)/Center,

PSp2m(K) = Sp2m(K)/Center,

PSUn(K) = SUn(K)/Center coming from the general linear, sympletic, and unitary groups respectively, and three others coming from the orthogonal group.

There Lie groups discussed represent infinite families of finite simple groups. There are some addition Lie type groups as called the exceptional and or twisted Lie type groups. The 26 Sporadic groups Mathieu groups M11, M12, M22, M23, M24 Janko groups J1, J2 or HJ, J3 or HJM, J4 Conway groups Co1 or F2, Co2, Co3 Fischer groups Fi22, Fi23, Fi24 or F3+ HigmanSims group HS McLaughlin group McL Held group He or F7+ or F7 Rudvalis group Ru Suzuki Suz or F3 O’Nan group O’N HaradaNorton group HN or F5+ or F5 Lyons group Ly Thompson group Th or F3—3 or F3 Baby B or F2+ or F2 FischerGriess Monster group M or F1

From http://en.wikipedia.org/wiki/Sporadic group. The Monster Group The Monster group has order 808017424794512875886459904961710757005754368000000000. In a paper title: “Fake Baby Monster Lie Algebra.”

http://en.wikipedia.org/wiki/Sporadic group. References “The Status of the Classification of the Finite Simple Groups” by in Notices of the AMS Volume 51, Number 7.

“Finite Simple Groups” by Hurley and Rudvalis, The American Mathematical Monthly Volume 84, Number 9 (1977).

Wikipedia, particularly http://en.wikipedia.org/wiki/Sporadic group.

“A first course in Abstract Algebra,” third edition, by Rotman. Published by Prentice Hall 2006.

Simplicity of the alternating group. http://planetmath.org/encyclopedia/SimplicityOfA n.html

The Alternating Group is Simple. people.reed.edu/ jerry/332/08ansim.pdf.