Commutator of finite nilpotent group with coprime automorphism group equals second commutator
From Groupprops
Statement
Let be a Finite nilpotent group (?) and be a subgroup of whose order is relatively prime to the order of . Then, we have:
,
where:
.
Related facts
- Commutator of finite group with cyclic coprime automorphism group equals second commutator
- Commutator of finite group with coprime automorphism group equals second commutator
- Centralizer-commutator product decomposition for abelian groups
- Centralizer-commutator product decomposition for finite nilpotent groups
- Centralizer-commutator product decomposition for finite groups and cyclic automorphism group
- Centralizer-commutator product decomposition for finite groups
Facts used
- Centralizer-commutator product decomposition for finite nilpotent groups: This states that under the same conditions ( a finite nilpotent group and a subgroup of of order coprime to the order of ), we have , and further, if is an -invariant subgroup of such that , then .
- Nilpotence is subgroup-closed
- Lagrange's theorem
- Order of quotient group divides order of group
Proof
Given: A finite nilpotent group, a subgroup of such that the orders of and are relatively prime.
To prove: .
Proof: Let .
- : Since , we have , so . Thus, , so . Thus, .
- : This follows directly from fact (1).
- : This follows by applying fact (1), replacing by , and observing that:
- Since is nilpotent, fact (2) tells us that is nilpotent.
- Since has order relatively prime to that of , the order of is also relatively prime to (using fact (3)). Further, since is -invariant, there is a homomorphism , with image of order dividing the order of , and so that the action of on factors through this homomorphism. The order of divides the order of by fact (4), so we get the conditions for fact (1), yielding . Finally, since the action of factors via the homomorphism, we get .
- : By the two preceding steps, . But , so we get .
- is -invariant: Since , we in particular have that is invariant.
- : By the preceding two steps, is -invariant and , so by fact (1), we have .
- : This follows from the preceding step, combined with step (1).
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 181, Theorem 3.6, Section 5.3 (p'-automorphisms of p-groups), ^{More info}