UNIVERSIDADE FEDERAL DE MINAS GERAIS INSTITUTO DE CIENCIAS^ EXATAS DEPARTAMENTO DE MATEMATICA´ Frederico Augusto Menezes Ribeiro Finite p-groups with three characteristic subgroups and a class of simple anticommutative algebras BELO HORIZONTE October, 2016 Frederico Augusto Menezes Ribeiro Finite p-groups with three characteristic subgroups and a class of simple anticommutative algebras Thesis submitted to Universidade Federal de Minas Gerais for the degree of Doctor of Philosophy in Mathematics Advisor: Csaba Schneider Belo Horizonte October, 2016 Abstract The structure of finite characteristically simple groups is very well known. One might then ask what happens when a group has few characteristic subgroups. UCS groups are one step from characteristically simple groups into the direction of more general groups: UCS groups are groups that have precisely three characteristic sub- groups. These groups however are way more complicated than characteristically simple groups. In this work we take a deeper look at a subclass of UCS groups, and how this subclass is related to the class of irreducible anticommutative algebras (IAC). We prove that there is a bijection between the set of isomorphism classes of finite- dimensional IAC algebras over the field with p elements and the set of isomorphism classes of finite UCS p-groups of exponent p2. From this bijection we show how to derive some properties of the group by looking at the corresponding algebra, showing for example that a group has proper nontrivial powerfully embeded subgroups if and only if the corresponding algebra has proper nontrivial ideals. We also investigate the structure of the IAC algebras showing that they are always semisimple (and actually always a direct sum of copies of a simple IAC algebra). In the last chapter we provide, by using three different methods, examples of infinite families of simple IAC algebras. i Abstract A estrutura de grupos simples caracter´ısticamente simples ´ebastante bem conhecida. E´ natural ent~aoperguntar-se o que ocorre quando um grupo tem poucos subgru- pos caracter´ısticos. Grupos UCS est~aoa um passo dos grupos caracter´ısticamente simples na dire¸c~aode grupos mais gerais: Grupos UCS s~aoos grupos que possuem precisamente tr^essubgrupos caracter´ısticos. Estes grupos por´ems~aobem mais com- plicados do que grupos caracter´ısticamente simples. Neste trabalho observamos mais precisamente uma subclasse dos grupos UCS, e como esta subclasse est´arelacionada com a classe de ´algebrasirredut´ıveis e anticomutativas (IAC). N´osprovamos que existe uma bije¸c~aoentre o conjunto das classes de isomorfismo de ´algebrasIAC de dimens~aofinita sobre o corpo como p elementos e o conjunto de classes de isomorfismo de p-grupos finitos UCS de expoente p2. A partir desta bije¸c~aon´osmostramos como obter algumas propriedades do grupo ao olhar para a ´algebracorrespondente, mostrando por exemplo que um grupo tem subgrupos powerfully embeded n~aotriviais pr´opriosse e somente se a ´algebracorrespondente tem ideais n~aotriviais pr´oprios. Tamb´eminvestigamos a estrutura das ´algebraIAC, mostrando que elas s~aosem- pre semissimples (e na verdade sempre s~aosoma direta de c´opiasde uma ´algebra IAC simples). No ´ultimo cap´ıtulon´osapresentamos, usando tr^esm´etodos distintos, exemplos e fam´ıliasinfinitas de ´algebrasIAC simples. ii Acknowledgements To my parents for everything. To my brother for the inspiration. To my advisor Csaba Schneider, and to Stephen Glasby for the patience. To my family and friends for the support. To FAPEMIG and Capes for the financial support. iii Contents Abstracti Abstract ii Acknowledgements iii 1 Preliminaries1 1.1 Introduction................................1 1.2 Groups...................................4 1.3 Finite p-groups..............................7 1.4 UCS groups................................ 13 1.5 Earlier results about UCS groups.................... 14 1.6 Tensor and exterior products....................... 18 1.7 Wreath Product.............................. 19 2 UCS p-groups and IAC algebras 22 2.1 Bijection between UCS groups and IAC algebras............ 22 2.2 Automorphisms of UCS groups..................... 30 2.3 ESQ modules............................... 32 iv 3 Structure of IAC algebras 34 3.1 Direct powers of UCS p-groups..................... 34 3.2 Direct sums of IAC algebras....................... 35 3.3 Normal subgroups and ideals....................... 38 3.4 nCS groups................................ 40 3.5 Abelian nCS Groups........................... 42 4 Examples 46 4.1 IAC Algebras of dimension 3....................... 46 4.2 A family of IAC algebras with an irreducible cyclic group....... 49 4.3 Another family of IAC algebras..................... 52 4.4 The Clebsch-Gordan formula for the exterior product......... 56 v Chapter 1 Preliminaries 1.1 Introduction Sylow's theorem states that a finite group always contains p-subgroups of order equal to the largest power of p dividing the order of the group. The structure of these subgroups does have an impact on the structure of the bigger group. Thus understanding the theory of p-groups helps us with the general theory of groups. One problem that arises with this approach is that the number of p-groups is very large. For example, there are 56092 groups of order 28, 10494213 groups of order 29, and 49487365422 groups of order 210 (see [2, Table 1]). It is natural to study subsets of p-groups with some property (see for example [3] which studies p-groups of maximal nilpotency class for their order). One method used to study not only p-groups, but groups in general is considering their groups of automorphisms, see for example [13] where by considering a subgroup of Aut(G) the authors of the paper determine an upper bound for the exponent of G. One might then ask what happens to groups where the automorphisms do not leave any non-trivial proper subgroups fixed, that is, characteristically simple groups. A finite group is characteristically simple if and only if it is the direct product of copies of a simple group. A group G is said to be UCS (Unique Characteristic Subgroup) if it has precisely three characteristic subgroups. Since the subgroups 1 and G are always character- istic, a group G is UCS when it has exactly one nontrivial proper characteristic subgroup. Taunt in [26, page 1] says \It was suggested to the author by Prof. Hall that finite groups possessing exactly one proper characteristic subgroup would repay attention". 1 Taunt in [26] divided the study of UCS groups between solvable and nonsolvable groups, and noticed that finite solvable UCS groups have order divisible by at most two different primes, whereas for finite nonsolvable UCS groups the order is always divisible by at least three primes. In that paper he focused on finite UCS groups whose order is divisible by two primes. He promised a following paper dealing with UCS p-groups, but never delivered. In [8] Glasby, P´alfyand Schneider studied finite UCS p-groups, observing that this study could be done by investigating the Aut(G)-modules Φ(G) and G=Φ(G) where Φ(G) denotes the Frattini subgroup of G (See [8, Theorem 5]). They divided the class of these groups into the groups of exponent p, and the groups of exponent p2, using distinct methods in each case. For groups of exponent p2, the quotient V = G=Φ(G) has the property that the exterior product V ^V contains a submodule U such that (V ^ V )=U is isomorphic to V . They called such a module an ESQ (Exterior Self Quotient) module. The study in [8, Section 7] of the UCS p-groups of exponent p2 was done through the investigation of finite groups that admit an irreducible ESQ module of low dimension. In this work we will deal with these ESQ modules, with a natural product that comes from their ESQ structure, as nonassociative algebras whose group of auto- morphisms acts irreducibly on them. We call these algebras IAC. IAC algebras are not necessarily Lie, Malcev, alternative or Jordan algebras. We show the following theorem. Theorem 1.1.1 There is a bijection G from the set of isomorphism classes of IAC algebras of finite dimension over Fp to the set of isomorphism classes of finite UCS p-groups of exponent p2. We denote the inverse of the bijection G by L. Given a UCS p-group G of exponent p2, we consider the structure of L(G) to discover some properties of G. We prove the following theorem that shows that the existence of powerfully embeded proper non-trivial subgroups in G is linked to the existence of ideals in L(G). Theorem 1.1.2 Let G be a finite UCS p-group of exponent p2 and L = L(G). Then there exists a bijection β between the set of powerful subgroups of G that contain Φ(G) and the set of subalgebras of L. Further, the image by β of the set of powerfully embedded subgroups of G containing Φ(G) is the set of ideals of L. The idea of considering another algebraic structure to learn about a group is not new. We have for example the relation between groups and Lie rings which Khukhro 2 described in his book [12, page xiv]: \The Lie ring method of solving group-theoretic problems consists of three major steps. First, the problem must be translated into a corresponding problem about Lie rings constructed from the groups. Then the Lie ring problem is solved. The results on Lie rings must then be translated back, into required conclusions about groups." This is what we will do with UCS p-groups of exponent p2 and IAC algebras. As was remarked by Schafer in [22, page 6] the study of both Lie and associative algebras is done by finding a radical such that the algebra modulo this radical is semisimple. By following the same procedure for IAC algebras, we have proved in this work the following theorem.