Groupprops, The Group Properties Wiki (pre-alpha)
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From Groupprops

Contents 1 Statement 2 Related facts 3 Facts used 4 Proof

Statement

Suppose G is a group with center Z(G), inner automorphism group , and automorphism group . Suppose the following two statements are true:

• Every automorphism of G is a center-fixing automorphism: it fixes every element of the center Z(G).
• is a maximal subgroup of .

Then, every automorphism of G is a normal-extensible automorphism: whenever G is a normal subgroup of some group K and σ is an automorphism of G, σ extends to an automorphism σ' of K.

Related facts

• Centerless and maximal in automorphism group implies every automorphism is normal-extensible
• Finite p-group with center of prime order and inner automorphism group maximal in p-Sylow-closure of automorphism group implies every p-automorphism is p-normal-extensible
• Every automorphism is center-fixing and outer automorphism group is rank one p-group implies not every normal-extensible automorphism is inner

Facts used

1. Center-fixing implies central factor-extensible: If σ is a center-fixing automorphism of a central factor G of a group K, σ extends to an automorphism of K.
2. Equivalence of definitions of central factor

Proof

Given: A group G such that every automorphism of G fixes every element of Z(G), and is a maximal subgroup of .

To prove: For any automorphism σ of G, and any group K containing G as a normal subgroup, σ extends to an automorphism of K.

Proof: Let be the homomorphism given by the action of K on G by conjugation. This map exists because G is normal in K.

1. The image of α is either or : The image of α is a subgroup of . It contains , because equals α(G). Since is a maximal subgroup of , the image is either or .
2. If the image is , then every automorphism of G extends to an inner automorphism of K, and we are done. (We say in this case that G is a normal fully normalized subgroup of K).
3. If the image is , then G is a central factor of K, i.e., K = GC_K(G) (see fact (2)). Thus, by fact (1), any automorphism σ that fixes the center of G extends to an automorphism of K. Since we assumed that every automorphism fixes the center of G, we obtain that every automorphism of G extends to an automorphism of K.