Varietal isologism and covering groups
Let \(\mathcal V\) be a variety of groups defined by the set \(V\) of laws and \(G\) a group. Then \(V(G)\) and \(V^*(G)\) stands for, respectively, the verbal subgroup and the marginal subgroup associated with \(\mathcal V\) and \(G\). If \(G\) and \(H\) are two groups then \((\alpha,\beta)\) is said to be a \(\mathcal V\)-isologism between \(G\) and \(H\) if \(\alpha\colon G/V^*(G)\to H/V^*(H)\) and \(\beta\colon V(G)\to V(H)\) are isomorphisms such that, for all \(v(x_1,\dots,x_r)\in V\) and all \(g_1,\dots,g_r\in G\), we have \(\beta(v(g_1,\dots,g_r))=v(h_1,\dots,h_r)\) whenever \(h_i\in\alpha(g_iV^*(G))\), \(i=1,\dots,r\). The Baer-invariant of \(G\) with respect to the variety \(\mathcal V\) is denoted by \({\mathcal V}M(G)\). Let \(1\to A\to G^*\to G\to 1\) be an exact sequence. If \(A\subseteq V(G^*)\cap V^*(G^*)\) and \(A\cong{\mathcal V}M(G)\) then \(G^*\) is called a \(\mathcal V\)-covering group of \(G\). It is proved that any two \(\mathcal V\)-covering groups of a given group are \(\mathcal V\)-isologic (Theorem 2.3). Theorem 3.2 states that every automorphism of a finite \(\mathcal V\)-perfect group \(G\) can be lifted to an automorphism of its \(\mathcal V\)-covering group.
scientific article; zbMATH DE number 1507820