The group of automorphisms of an elementary-abelian-over-cyclic regular wreath product \(p\)-group. (English) Zbl 1472.20047
Summary: Let \(W\) denote the regular wreath product finite group \(C \wr E\) where \(C\) is a cyclic \(p\)-group and \(E\) is an elementary abelian \(p\)-group. Let \(A\) denote the subgroup of \(\operatorname{Aut}(W)\) consisting of those automorphisms that act trivially on \(W/B\), where \(B\) is the base group. We determine \(A\) by describing where each of its elements map a certain generating set for \(W\). We find that \(A\) is as large as possible in a certain sense. We determine some information about the subgroup structure of \(A\), and we prove that every class-preserving automorphism of \(W\) is an inner automorphism of \(W\).
MSC:
20D45 | Automorphisms of abstract finite groups |
20D15 | Finite nilpotent groups, \(p\)-groups |
20E22 | Extensions, wreath products, and other compositions of groups |
20F28 | Automorphism groups of groups |
References:
[1] | Hertweck, M., Contributions to the integral representation theory of groups, Habilitationsschrift (2004), University of Stuttgart |
[2] | Houghton, C. H., On the automorphism groups of certain wreath products, Publ. Math. Debrecen, 9, 307-313 (1962) · Zbl 0118.26702 |
[3] | Liebeck, H.; Taunt, D. R., Concerning nilpotent wreath products, Math. Proc. Camb. Phil. Soc, 58, 3, 443-451 (1962) · Zbl 0106.24902 · doi:10.1017/S0305004100036719 |
[4] | Neumann, P. M., On the structure of standard wreath products of groups, Math. Z, 84, 4, 343-373 (1964) · Zbl 0122.02901 · doi:10.1007/BF01109904 |
[5] | Riedl, J. M., Automorphisms of regular wreath product p-groups, Int. J. Math. Math. Sci, 2009 (2009) · Zbl 1189.20025 · doi:10.1155/2009/245617 |
[6] | Riedl, J. M., Commutator calculus for wreath product groups, Comm. Algebra, 43, 5, 2152-2173 (2015) · Zbl 1328.20051 · doi:10.1080/00927872.2014.888564 |
[7] | Riedl, J. M., Upper central series for elementary-abelian-over-cyclic regular wreath product p-groups, J. Algebra Appl, 14, 4 (2015) · Zbl 1326.20033 · doi:10.1142/S0219498815500449 |
[8] | Yadav, M. K., On automorphisms of some finite p-groups, Proc. Math. Sci, 118, 1, 1-11 (2008) · Zbl 1148.20014 |
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