# Centralizer-commutator product decompoition for abelian groups

## Statement

Suppose is an Abelian group and is a finite group such that the map is a bijective map on (when is finite, this is equivalent to requiring that the orders of and be relatively prime). Define:

- is the subgroup generated by elements of of the form , where .
- is the subgroup comprising those such that .

Note that these correspond to the usual notions of commutator of two subgroups and centralizer if we look inside the semidirect product of by .

Then, is the internal direct product of the subgroups and . In other words, we have:

.

## Related facts

### Other averaging lemmas

- Maschke's averaging lemma for modules: This is the most general form, stating that if a group acts on a module over a ring in which its order is invertible, and there is an invariant direct summand of the module, the direct summand has an invariant complement.
- Maschke's averaging lemma for Abelian groups: A somewhat weaker reformulation of the above.
- Maschke's averaging lemma: A variant on the lemma for modules, where the base ring is a field.

### Applications

- Centralizer-commutator product decomposition for finite nilpotent groups
- Centralizer-commutator product decomposition for finite groups

## Proof

**Given**: A finite Abelian group , a group such that the orders of and are relatively prime.

**To prove**:

**Proof**: We define a map , prove that is a projection, and show that the kernel and image of are and respectively.

Define:

.

Note that this expression makes sense because the multiplication by map is invertible in by the condition of relatively prime orders.

We observe that:

.

This is because and yield the same summation as , in a different order, and the group is Abelian.

This further yields:

.

Thus, is an idempotent endomorphism of . In other words, the image of equals its fixed-point space.

- Computation of the image of :
- is in the image: If , then for all , so . Thus, , so is in the fixed-point space of .
- The image is in : If is fixed under , then , so is fixed by all . Thus, the fixed-point space of , which is also the image of , is precisely .

- Computation of the kernel of :
- is in the kernel: Let's now consider the kernel of . Suppose for some . Then . Thus, any element in is in the kernel of .
- The kernel is in : If , then . This yields that . Thus, the kernel of is precisely .