# Group Theory/Characteristic subgroups

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**Definition (characteristic subgroup)**:

Let be a group. A **characteristic subgroup** of is a subgroup such that for all .

**Proposition (characteristic subgroups are normal)**:

Any characteristic subgroup of a group is a normal subgroup of .

**Proof:** This follows since the map is a group automorphism of .

**Definition (characteristically simple)**:

A group is called **characteristically simple** if and only if its only characteristic subgroups are and , where denotes the identity of .

**Proposition (characteristically simple groups)**:

Let be a characteristically simple finite group, and let be any of its minimal normal subgroups. Then is isomorphic to a product of copies of , that is, , where is an index set (of finite cardinality).

**Proof:** Let be a subgroup of maximal subject to the following two conditions:

- is the direct sum of images of under a
- is normal

Suppose that . Note that the group is characteristic, so that it equals all of . Hence, we find such that is not a subgroup of . Since is an automorphism, , so that . Since the product of normal subgroups is normal, we conclude that the product subgroup is a normal subgroup that is a direct product of homomorphic images of in , in contradiction to the maximality of with these properties. Hence, and we are done.

**Proposition (minimal normal subgroups of a characteristically simple groups are simple)**:

Let be a characteristically simple group, and let be a minimal normal subgroup of . Then is simple.

**Proof:**

**Proposition (powers of characteristically simple groups are characteristically simple)**:

We conclude:

**Theorem (structure theorem of finite, characteristically simple groups)**:

The finite, characteristically simple groups are precisely the powers of simple groups.

**Proof:** We have seen that each characteristically simple finite group is the direct product of copies of isomorphic images of any of its minimal normal subgroups, and that the latter are always simple in characteristically simple groups. We conclude that each finite, characteristically simple group is a power of simple groups. Conversely, let be a simple group, , and set

- .

## Exercises

edit- Prove that all subgroups of are characteristic.
- Let be two finite simple groups such that is divisible by a prime number that does not divide . Use the structure theorem for characteristically simple groups to prove that is not characteristically simple.
- Prove that a subgroup of a characteristically simple group need not be characteristically simple.
- Prove that the product of characteristically simple subgroups whose minimal normal subgroups are not isomorphic is not characteristically simple.