Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

Contents 1 Statement 2 Related facts 2.1 Applications 3 Facts used 4 Proof

Statement

Suppose F is a prime field (i.e., either a field of prime order or the field of rational numbers), and G is a group. Suppose V is a finite-dimensional vector space over F, and be a linear representation of G. Let with respect to the induced action of G on V.

Suppose, further, that σ is an automorphism of G that can be extended to an automorphism σ' of H such that σ' also restricts to an automorphism α of V. Then, where c_α is conjugation by α in GL(V).

Note that we need the field to be a prime field in order that GL(V) is equal to the automorphism group of V as a group.

Related facts

• Quotient-pullbackable implies linearly pushforwardable for representation over prime field

Applications

• Finite-extensible implies class-preserving
• Hall-semidirectly extensible implies class-preserving
• Finite solvable-extensible implies class-preserving
• Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving
• Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving

Facts used

1. Automorphism group equals general linear group for vector space over prime field
2. Automorphism group action lemma: Suppose H is a group, and are subgroups such that . Suppose σ' is an automorphism of H such that the restriction of σ' to N gives an automorphism α of N, and such that σ' also restricts to an automorphism of G, say σ. Consider the map:

that sends an element to the automorphism of N induced by conjugation by g (note that this is an automorphism since ). Then, we have:

where c_α denotes conjugation by α in the group .

Proof

Given: A group G, a homomorphism for a finite-dimensional vector space V over a prime field F. σ is an automorphism of G that extends to an automorphism σ' of H, such that σ' also restricts to an automorphism α of V.

To prove: .

Proof: Since F is a prime field, GL(V) is the whole automorphism group of V by fact (1) (in general, it is a proper subgroup). Thus, the element α, which is a group automorphism of V, is actually in GL(V). Thus, fact (2), setting G = G,H = H,N = V,σ' = σ',α = α,σ = σ, gives the desired result.