Centralizer-commutator product decomposition for finite nilpotent groups
Suppose is a finite nilpotent group and is such that the orders of are relatively prime. Define:
- as the subgroup generated by all elements of the form where .
- as the subgroup of comprising those such that for all .
Then, we have the following:
Further, if is a -invariant subgroup such that , then .
- Stability group of subnormal series of finite group has no other prime factors
- Centralizer-commutator product decomposition for Abelian groups: This states that the result holds when is Abelian (in fact, something stronger holds: is an internal direct product of the two subgroups).
- Commutator of a group and a subgroup of its automorphism group is normal
- Nilpotent implies center is normality-large
- Nilpotence is quotient-closed
Given: A finite nilpotent group , a subgroup of such that the orders of and are relatively prime.
To prove: and if , then .
Proof: We first prove that if , then :
Pick any . We want to show that .
Since , we have , where . Thus, we have:
Since , , so we get:
Since and is -invariant, , so , completing the proof.
We now prove that .
We first consider the case that is contained in the center of .
- The map is an endomorphism of for any : For any , we have . Since , we can commute with to get . This proves the condition for being a homomorphism.
- contains the subgroup : Note that can be described as the intersection of kernels of for all . Each has image inside the center of by definition, hence is a map to an Abelian group. Thus, the kernel of each contains the commutator subgroup . Hence, the intersection, which is , also contains .
- Let and be the quotient map. Then, acts on by the natural induced action, , and :
- Observe that since is characteristic in , it is -invariant, so acts on the quotient group. Note that this descent satisfies for all .
- : This follows directly from the definition.
- Now, suppose . Clearly, if , then for all . Thus, , so . Thus, , so .
- Consider the subnormal series for . acts as the identity on since , and it acts as the identity on since . Thus, acts as stability automorphisms of this subnormal series, and its order is relatively prime to , and hence to . Fact (1) thus forces that acts as the identity on , so .
- Combining the previous two steps yields .
- We have : This follows from fact (2), and the observation that since the orders of and are relatively prime, so are the orders of and .
- : From steps (3) and (4), we have . Note that is a subgroup, since , hence is normal. Thus, is a subgroup of whose image via is the whole quotient . On the other hand, step (2) says that , so contains the kernel and intersects every coset of it -- hence must be equal to the whole group .
Case that is not contained in the center
- is normal in : This follows from fact (3).
- intersects the center nontrivially: This follows from fact (4), and the fact that is normal.
- Let . Then, is nontrivial, -invariant, and normal: is characteristic, and hence -invariant, while we have , so is -invariant. Thus, the intersection is invariant. Further, is normal since it is the intersection of the normal subgroups and (or alternatively, is a subgroup of ).
- Let , and be the quotient map. Then, the action of on descends to an action on : This follows from step (3), where it is observed that is normal and -invariant.
- For the induced action of on , we have : Since is the quotient of by a nontrivial subgroup, it has strictly smaller order. Since is nilpotent, so is (fact (5)). Thus, the induction hypothesis applies to .
- Let . Then, : Clearly, we have . Thus, , while also contains the kernel of . Hence, is a subgroup (it is one because is normal, step (1)) containing the kernel of and intersecting every coset of this kernel, forcing .
- : Note that since , we have which is trivial. Thus, , and we know that is in the center of , hence . Thus, , and the previous case kicks in for .
- : This follows by piecing together the previous two steps.
- : Since , we have . Also, . Substituting these gives the required result.