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| {{automorphism property implication|
| | #redirect [[Extensible implies subgroup-conjugating]] |
| stronger = extensible automorphism|
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| weaker = permutation-extensible automorphism}}
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| ==Statement==
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| Any [[extensible automorphism]] of a [[group]] is a [[permutation-extensible automorphism]].
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| ==Definitions used==
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| ===Extensible automorphism===
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| {{further|[[Extensible automorphism]]}}
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| An automorphism <math>\sigma</math> of a group <math>G</math> is termed '''extensible''' if, for any embedding of <math>G</math> in a bigger group <math>H</math>, there exists an automorphism <math>\sigma'</math> of <math>H</math> such that the restriction of <math>\sigma'</math> to <math>G</math> equals <math>\sigma</math>.
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| ===Permutation-extensible automorphism===
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| {{further|[[Permutation-extensible automorphism]]}}
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| An automorphism <math>\sigma</math> of a group <math>G</math> is termed '''permutation-extensible''' if, for any embedding of <math>G</math> in a symmetric group <math>\operatorname{Sym}(S)</math>, there exists an element <math>h \in \operatorname{Sym}(S)</math> such that if <math>\sigma' = c_h</math> is conjugation by <math>h</math>, the restriction of <math>\sigma'</math> to <math>G</math> is <math>\sigma</math>. In other words, <math>\sigma</math> extends to an inner automorphism of <math>\operatorname{Sym}(S)</math>.
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| ==Related facts==
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| ===Applications===
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| * [[Extensible implies subgroup-conjugating]]
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| * [[Extensible implies normal]]
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| * [[Extensible automorphism-invariant equals normal]]
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| ==Facts used==
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| # [[uses::Symmetric groups on finite sets are complete]]: For <math>n</math> a natural number other than <math>2</math> or <math>6</math>, the symmetric group on <math>n</math> elements is a complete group. In particular, every automorphism of it is inner.
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| # [[uses::Symmetric groups on infinite sets are complete]]: The symmetric group on any infinite set is a complete group. In particular, every automorphism of it is inner.
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| ==Proof==
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| '''Given''': A group <math>G</math>, an extensible automorphism <math>\sigma</math> of <math>G</math>. A set <math>S</math> with an embedding <math>G \to \operatorname{Sym}(S)</math>.
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| '''To prove''': <math>\sigma</math> extends to an inner automorphism of <math>\operatorname{Sym}(S)</math>.
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| '''Proof''': We consider the following cases:
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| * <math>S</math> is infinite: By assumption, <math>\sigma</math> extends to an automorphism of <math>\operatorname{Sym}(S)</math>. By fact (2), this automorphism must be inner. Hence, <math>\sigma</math> extends to an inner automorphism of <math>\operatorname{Sym}(S)</math>.
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| * <math>S</math> is finite, and its cardinality is different from <math>2</math> or <math>6</math>: By assumption, <math>\sigma</math> extends to an automorphism of <math>\operatorname{Sym}(S)</math>. By fact (2), this automorphism must be inner. Hence, <math>\sigma</math> extends to an inner automorphism of <math>\operatorname{Sym}(S)</math>.
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| * <math>S</math> is finite with cardinality <math>2</math>: By assumption, <math>\sigma</math> extends to an automorphism of <math>\operatorname{Sym}(S)</math>. But there's only one automorphism of the symmetric group on a two-element set: the identity automorphism. This is clearly inner, so we are done.
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| * <math>S</math> is finite with cardinality <math>6</math>: We consider two cases.
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| ** There is an element <math>s \in S</math> such that every element of <math>G</math> fixes <math>s</math>: In this case, <math>G</math> is a subgroup of the subgroup <math>\operatorname{Sym}(S \setminus \{ s \})</math>, which is the symmetric group on a set of size five. Since <math>\sigma</math> is extensible, it extends to an automorphism of <math>\operatorname{Sym}(S \setminus \{ s \})</math>, and by fact (1), this automorphism must be inner. This inner automorphism can further be extended to an inner automorphism of <math>\operatorname{Sym}(S)</math>, by using the same permutation.
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| ** There is no element of <math>S</math> fixed by all elements of <math>G</math>: Let <math>T = S \sqcup \{ x_0 \}</math> with <math>G</math> acting on <math>x_0</math> trivially. Thus, <math>G</math> acts on <math>T</math>, with <math>G \le \operatorname{Sym}(S) \le \operatorname{Sym}(T)</math>. <math>T</math> is a set of size seven. Since <math>\sigma</math> is extensible, it extends to an automorphism of <math>\operatorname{Sym}(T)</math>, and by fact (1), this automorphism must be inner. Suppose <math>h \in \operatorname{Sym}(T)</math> is a permutation giving this inner automorphism. Then, since <math>G</math> fixes <math>x_0</math>, <math>hGh^{-1}</math> fixes <math>hx_0</math>. Since <math>hGh^{-1} = \sigma(G) = G</math>, we get that <math>G</math> fixes <math>hx_0</math>. Since no element of <math>S</math> is fixed by the whole of <math>G</math>, <math>hx_0 = x_0</math>. Thus, the permutation <math>h</math> restricts to a permutation on the subset <math>S</math>, and this inner automorphism gives the required inner automorphism extending <math>\sigma</math> to <math>\operatorname{Sym}(S)</math>.
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