Realizing finite groups as automizers (Q6565826)

The authors study the realization of finite groups by automizers of subgroups of finite groups. That is, given a finite group \(A\), they study when it is possible to find a finite group \(G\) and a subgroup \(U \leq G\) such that \(A \cong {\Aut}_G(U) \cong N_G(U)/C_G(U)\). The answer to this question is \lq always possible\rq \ for trivial reasons: choose a faithful action of \(A\) on an elementary abelian \(p\)-group \(U\) and take for \(G\) the semidirect product of \(U\) by \(A\). In this case, \(U\) is normal in \(G\). The main result shows that it is possible to realize \(A\) as \({\Aut}_G(U)\), where \(U\) is very far from being normal. It is shown that any finite group \(A\) is realizable as the automizer in a finite perfect group \(G\) of an abelian subgroup whose conjugates generate \(G\). The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by \textit{S. Park} [Arch. Math. 94, No. 5, 405--410 (2010; Zbl 1243.20025); Proc. Am. Math. Soc. 144, No. 8, 3291--3294 (2016; Zbl 1367.20016)].