Centralizer-commutator product decomposition for finite groups and cyclic automorphism group

From Groupprops
Jump to: navigation, search

Statement

Suppose G is a finite group and H is a cyclic subgroup of \operatorname{Aut}(G). Let [G,H] be the subgroup of G generated by all elements of the form g\sigma(g)^{-1} for \sigma \in H. Let C_G(H) denote the subgroup of G comprising those elements fixed by every element of H. Then:

G = [G,H]C_G(H).

Related facts

Facts used

  1. Centralizer of coprime automorphism in homomorphic image equals image of centralizer: Suppose \varphi is an automorphism of G of order coprime to the order of G. Suppose N is a normal \varphi-invariant subgroup of G. Then if \pi:G \to G/N denotes the quotient map, we have \pi(C_G(\varphi)) = C_{G/N}(\varphi).

Proof

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

References

Textbook references

  • Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, Page 18, Corollary 1.6.4, (Proof uses theorem 1.6.2)More info