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Prof. Dr. Eric Jespers Science Faculty Mathematics Department Bachelor Paper II

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Prof. Dr. Eric Jespers Science Faculty Mathematics Department Bachelor Paper II Prof. Dr. Eric Jespers Science faculty Mathematics department Bachelor paper II 1 Voorwoord Dit is mijn tweede paper als eindproject van de bachelor in de wiskunde aan de Vrije Universiteit Brussel. In dit werk bestuderen wij eindige groepen G die minimaal niet nilpotent zijn in de volgende betekenis, elke echt deelgroep van G is nilpotent maar G zelf is dit niet. W. R. Scott bewees in [?] dat zulke groepen oplosbaar zijn en een product zijn van twee deelgroepen P en Q, waarbij P een cyclische Sylow p-deelgroep is en Q een normale Sylow q-deelgroep is; met p en q verschillende priemgetallen. Het hoofddoel van dit werk is om een volledig en gedetailleerd bewijs te geven. Als toepassing bestuderen wij eindige groepen die minimaal niet Abels zijn. Dit project is verwezenlijkt tijdens mijn Erasmusstudies aan de Universiteit van Granada en werd via teleclassing verdedigd aan de Universiteit van Murcia, waar mijn mijn pro- motor op sabbatical verbleef. Om lokale wiskundigen de kans te geven mijn verdediging bij te wonen is dit project in het Engles geschreven. Contents 1 Introduction This is my second paper to obtain the Bachelor of Mathematics at the University of Brussels. The subject are finite groups G that are minimal not nilpotent in the following meaning. Each proper subgroup of G is nilpotent but G itself is not. W.R. Scott proved in [?] that those groups are solvable and a product of two subgroups P and Q, with P a cyclic Sylow p-subgroup and G a normal Sylow q-subgroup, where p and q are distinct primes. The main objective is to give a complete and detailed prove. As application we study finite groups that are minimal not Abelian. The structure of this project is as follows. In section 2 we recall the necessary background on Sylow Theorems, nilpotent and solvable groups. Proofs will only be given for those results that are not covered in the courses [1], [2], [3] and [4]. We finish this section with proving Burnside’s and Hall’s Theorems. The former says that every group of order paqb is solvable and the later states that if for each prime divisor of the order of a group G there exists a subgroup H of G such that the index of this subgroup in G equals the maximal power of the prime then G is solvable. If H is a proper subset of a group G then this is denoted by H ⊂ G, otherwise we denoted it by H ⊆ G. 2 Preliminary concepts 2.1 Sylow Theorems The Sylow Theorems are a really important tool in the study of finite groups that give detailed information about the number of subgroups of fixed order that a given finite group contains. Throughout this section is G a finite group. A group G is said to be a p-group if the order of each element in G is a power of p with p prime. The order of a p-group G is a power of p and the subgroups are also p-groups. If H is a normal subgroup of a p-group G, then H and G/H are p-groups. The converse is true as well, i.e. if H and G/H are p-groups then G is a p-group. The proofs of the properties of p-groups can be found in chapter 6 of [?]. For every group G the center of G is defined as the set Z(G) = {z ∈ G | zg = gz, for all g ∈ G}. It is well known that if G is a nontrivial finite p-group, then the center Z(G) is nontrivial. If S is a subset of G the centralizer and the normalizer of S, are the subgroups CG(S) = {g ∈ G | sg = gs for all s ∈ S} en NG(S) = {g ∈ G | gS = Sg}. They are denoted in short as C(S) and N(S). An other special type of subgroups of a group G are maximal subgroups. They are so called because they are a maximal element of the partially ordered set of proper subgroups of G. This means that they are proper sub- groups, such that no proper subgroup K contains H strictly. A maximal element in the set of all p-subgroups of a group G is called a Sylow p-subgroup. The set consisting of the Sy- low p-subgroups is denoted by Sylp(G). The number of elements in Sylp(G) is denoted np(G). 1 Ludwig Sylow summarized all of this and more in his theorems. r Theorem 2.1 (Sylow). If G is a group of order p m, p prime and (p, m) = 1, P ∈ Sylp(G) then (a) np ≡ 1 (mod p), np | m, np(G) = [G : NG(P )], (b) all Sylow p-subgroups are conjugate, and (c) if H ⊂ G, then H is a Sylow-subgroup if and only if |H| = pr. Theorem 2.2. Let G be a finite group and P ∈ Syp(G). The following statements are equivalent: 1. P is a normal subgroup; 2. P is the only Sylow p-subgroup; 3. each p-subgroup is contained in P ; 4. P is characteristic in G, i.e. P is invariant under group automorphisms of G. Theorem 2.3. If |G| = prm, p prime, (p, m) = 1,H ⊂ G, and |H| = ps, then H is contained in some (Sylow) subgroup. Theorem 2.4. If H ⊂ G with G finite and p prime, then np(H) ≤ np(G). Proof. If H is a proper p-subgroup of G then there exists a maximal p-subgroup of G con- ∗ taining H. This implies that if P ∈ Sylp(H) then there exists a P ∈ Sylp(G) such that P ⊆ P ∗. Now we have that every Sylow p-subgroup of H is contained in a Sylow p-subgroup of G. It remains to prove that if P and Q are two distinct Sylow p-subgroups of H and P ⊆ P ∗,Q ⊆ Q∗ with P ∗,Q∗ Sylow p-subgroups of G then P ∗ 6= Q∗. If P ∗ = Q∗, then P ∗ ⊇ hP, Qi and thus is < P, Q > a p-subgroup of H such that hP, Qi ⊃ P , which contradicts P being maximal p-subgroup. 2.2 Solvable groups A solvable group (or soluble group) is a group that can be constructed from Abelian groups using extensions. A precise definition is as follows: Definition 2.5. A group G is solvable if there exist normal subgroups G0,G1,G2, ..., Gn of G such that 1. {1} = G0 ⊆ G1 ⊆ ... ⊆ Gn = G, 2. Gi+1/Gi is Abelian for 0 ≤ i < n. Some examples are Abelian groups, S3, D8 and all finite groups of prime order. The derived group G0 is the subgroup of G generated by the commutators [g, h] = g−1h−1gh, g, h ∈ G. This is the smallest normal subgroup of G such that G/G0 is Abelian. It is also denoted by [G, G]. The nth derived subgroup (for n > 1) G(n) = (G(n−1))0 of G is defined by [G(n−1),G(n−1)]. 2 Definition 2.6. The series G = G(0) ⊇ G(1) ⊇ G(2) ⊇ ... is called the derived series of G. Theorem 2.7. A group G is solvable if and only if G(n) = {1}, for some n. The smallest n wherefore G(n) = {1} is called the derived length, denoted by dl(G). Definition 2.8. Let G0,G1, ..., Gn be subgroups of a group G such that {1} = G0 ⊆ G1 ⊆ G2 ⊆ ... ⊆ Gn−1 ⊆ Gn = G. This series is said to be a composition series of G if Gi C Gi+1 and Gi+1/Gi is simple for all 0 ≤ i ≤ n − 1. It is well known that a finite group has a composition series and, up to a permutation and isomorphism, the composition factors Gi+1/Gi are unique. Theorem 2.9. A group G is solvable if and only if each compositionfactor of G is a group of prime order. Theorem 2.10. If N is a normal subgroup of G then, G is solvable if and only if N and G/N are solvable. Definition 2.11. A solvable group is called supersolvable if G has a composition series con- sisting of normal subgroups in G. Definition 2.12. A group G is called metacyclic if there exists a normal subgroup N of G such that both N and G/N are cyclic. Examples of supersolvable groups are metacyclic groups and nilpotent groups. An example n of metacyclic groups are dihedral groups, D2n. Those latter groups are defined by < σ, τ|σ = τ 2 = 1, στ = τσ−1 >. 2.3 Nilpotent groups Nilpotency is a stronger concept then solvability. The following definition defines finite nilpo- tent groups by means of Sylow p-subgroups. Definition 2.13. A finite group G is called nilpotent if for every prime divisor p of the order of the group, the group G contains a normal Sylow p-subgroup. Theorem 2.14. If G is a finite nilpotent group, then Y G = Sylp(G), p a direct product of Sylow subgroups. The concept of nilpotent groups can also be defined for a arbitrary group. Definition 2.15. A group H is said to be nilpotent if there exist normal subgroups H0,H1, ..., Hn such that 1. {1} = H0 ⊆ H1 ⊆ ... ⊆ Hn = H, and 2. Hi+1/Hi ⊆ Z(H/Hi), for 0 ≤ i ≤ n − 1. Examples for nilpotent groups are Abelian groups and D8. An example of a solvable not nilpotent group is S3. 3 Definition 2.16.
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