Finite characteristically simple group

Definition

A group is termed a finite characteristically simple group if it satisfies the following equivalent conditions:

1. It is both a finite group and a characteristically simple group, i.e., it is a finite nontrivial group with no proper nontrivial characteristic subgroup.
2. It is nontrivial and can be expressed as an internal direct product of pairwise isomorphic finite simple groups.

Equivalence of definitions

Further information: equivalence of definitions of finite characteristically simple group

Two types

Abelian case

The finite characteristically simple abelian groups are precisely the finite elementary abelian groups. An elementary abelian group of order $p^{n}$ for a prime number $p$ can be thought of as a $n$-dimensional vector space over the field of $p$ elements. There are many different ways of describing this as an internal direct product of $n$ subgroups of order $p$, or equivalently, as a direct sum of $n$ one-dimensional subspaces. In fact, the number of such ways is $|GL(n,p)|/(n!(p-1)^{n})$: $|GL(n,p)|$ is the number of ordered bases. Division by $n!$ is to indicate that we are interested in unordered bases, not ordered bases. Division by $(p-1)^{n}$ is to indicate that we don't actually care about the individual elements, only about the one-dimensional subspace generated by that element.

Non-abelian case

The finite characteristicaly simple non-abelian groups are precisely the groups obtained as a finite direct power of a simple non-abelian group. In the non-abelian case, the internal direct product decomposition is unique making it quite different from the abelian case.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite simple group				|FULL LIST, MORE INFO