Centralizer-commutator product decompoition for abelian groups
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 .
Then, is the internal direct product of the subgroups and . In other words, we have:
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.
- Centralizer-commutator product decomposition for finite nilpotent groups
- Centralizer-commutator product decomposition for finite groups
Given: A finite Abelian group , a group such that the orders of and are relatively prime.
Proof: We define a map , prove that is a projection, and show that the kernel and image of are and respectively.
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 .