Outside in

Given a group G, embed G in a larger group H such that any automorphism of G is the restriction of an inner automorphism of H.

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Outside in

One thought on “Outside in

  1. Consider the semidirect product G\rtimes\mathrm{Aut}(G), which is called the holomorph of G and denoted \mathrm{Hol}(G). Obviously we have a copy of G in \mathrm{Hol}(G) and the product in \mathrm{Hol}(G) is such that \varphi g \varphi^{-1} = \varphi(g); in other words, in the holomorph, conjugation of a group element by an automorphism amounts to evaluation.

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