Finite p-groups with three characteristic subgroups and a class of simple anticommutative algebras

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 subgroups. 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 finitedimensionalIAC algebras over the eld with p elements and the set of isomorphismclasses 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 dierent methods, examples of infinite families of simple IAC algebras.