Finite-extensible implies inner

From Groupprops
Jump to: navigation, search
This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., finite-extensible automorphism) must also satisfy the second automorphism property (i.e., inner automorphism)
View all automorphism property implications | View all automorphism property non-implications
Get more facts about finite-extensible automorphism|Get more facts about inner automorphism


Suppose H is a finite group and \sigma is a finite-extensible automorphism of H: in other words, \sigma extends to an automorphism of G for any finite group G containing H. Then, \sigma is an inner automorphism of H.

Note that since any inner automorphism is extensible, this says that the property of being finite-extensible is equivalent to the property of being inner for a finite group.

Related facts

Weaker facts

Facts used

  1. Every finite group is the Fitting quotient of a p-dominated group for any prime p not dividing its order: Suppose H is a finite group and p is a prime not dividing the order of H. Then, there exists a p-dominated group G with H as Fitting quotient: in other words, there exists a finite complete group G such that the Fitting subgroup F(G) is a p-group, and H is a subgroup of G such that G = F(G) \rtimes H.


Given: A finite group H, a finite-extensible automorphism \sigma of H.

To prove: \sigma is inner.

Proof: Let p be a prime not dividing the order of H. Consider the group G constructed by fact (1). Since \sigma is finite-extensible, \sigma extends to an automorphism \sigma' of G. Further, since G is complete, there exists g \in G such that \sigma' is conjugation by g.

Let \rho:G \to H be the retraction with kernel F(G). Note that conjugation by g preserves F(G), hence it induces a conjugation map on H as a quotient, namely, conjugation by the element \rho(g) \in H. However, since the restriction of \rho to the subgroup H is the identity map, we conclude that conjugation by g has the same effect on H as conjugation by \rho(g). In particular, \sigma equals conjugation by \rho(g), and hence is inner.


Journal references