Groups with trivial virtual automorphism group
The authors consider groups G with the following property: Let H,K be subgroups of finite index in G and \(\phi: H\to K\) an isomorphism. Then there exists a subgroup L of \(H\cap K\) of finite index in G such that \(\phi\) is the identity on L. A finite group has this property. Mann and Lubotzky asked whether there exist infinite groups with this property. The authors answer this question by giving two examples of infinite groups having this property. The first example is an extraspecial 2-group of cardinality \(2^{\aleph_ 0}\) and the second example is a 2-group again of cardinality \(2^{\aleph_ 0}\) with derived subgroup isomorphic to a 4-group. It is an open problem whether there exists a group of countable cardinality with this property. They use a result of \textit{F. Menegazzo} and \textit{D. J. S. Robinson} [Rend. Semin. Mat. Univ. Padova 78, 267-277 (1987; Zbl 0637.20017)] and prove first that such a group G is virtually a 2-group with finite central derived subgroup and \(G/G''\) elementary abelian.
scientific article; zbMATH DE number 4190019