# Classification of fully characteristic subgroups in finitely generated Abelian groups

## Contents

## Statement

### Case of groups of prime power order

Suppose is an Abelian group whose order is a power of a prime . Then, we can write, by the structure theorem for finitely generated Abelian groups:

where:

.

where .

Then a subgroup of is a characteristic subgroup of if and only if is a fully characteristic subgroup of , and this happens if and only if the following are satisfied:

- is the direct sum of the intersections .
- If the orders of are , then and .

### Case of finite groups

Suppose is a finite Abelian group. Then, we can write:

where each is an Abelian group of prime power order for different primes. Then, a subgroup of is a characteristic subgroup if and only if it is a fully characteristic subgroup, if and only if the following are satisfied:

- is the direct sum of the intersections .
- Each is characteristic (equivalently, fully characteristic) in .

### Case of finitely generated groups

Suppose is a finitely generated Abelian group. Then, we can write:

where is a torsion-free group. A subgroup of is a fully characteristic subgroup of if and only if and we have that is fully characteristic in and is characteristic in .

## Proof

### Case of prime power order

**Given**: is an Abelian group whose order is a power of a prime . Then, we can write, by the structure theorem for finitely generated Abelian groups:

where:

.

where .

A subgroup of .

**To prove**: is characteristic if and only if it is fully characteristic in , if and only if is the direct sum of , and if the orders of the intersections are , then and .

**Proof**: The proof relies on two important homomorphisms. For , there is an injective homomorphism:

that sends the generator on the left to times the generator on the right.

There is also a surjective homomorphism:

that sends the generator to the generator.

Recall that we have:

.

For , define as the endomorphism of that sends to via the map:

.

and is zero elsewhere.

Define as the sum of the endomorphism and the identity map on . Clearly, is an automorphism.

Similarly, define as the endomorphism of that sends to via the map and is zero elsewhere. Define as the automorphism obtained as the sum of and the identity map.

We are now in a position to prove the main result.

Suppose is a characteristic subgroup of . We first show that is a direct sum of . For this, suppose the element:

.

We want to show that the elements are in .

If all except one are zero, then we are done. Otherwise, there exist such that both and are nonzero.