A note on the exterior centralizer.
If \(G\) is a group, the exterior square of \(G\) is the group \(G\wedge G\) generated by the symbols \(g\wedge h\) and subject to the relations \(g\wedge g=1\), \(g_1g_2\wedge h=(g_1g_2g_1^{-1}\wedge g_1hg_1^{-1})(g_1\wedge h)\) and \(g\wedge h_1h_2=(g\wedge h_1)(h_1gh_1^{-1}\wedge h_1h_2h_1^{-1})\) for all \(g,g_1,g_2,h,h_1,h_2\in G\). The kernel of the map \(\kappa\colon G\wedge G\to G'\), \(g\wedge h\mapsto [g,h]\) is naturally isomorphic to \(M(G)\), the Schur multiplier of \(G\). In the paper under review, the authors define the exterior centralizer of an element \(x\in G\) by \(C^{^{\wedge}}_G(x)=\{g\in G\mid g\wedge x=1\;\forall g\in G\}\) and prove that \(C^{^{\wedge}}_G(x)\) is a normal subgroup of \(C_G(x)\) and that \(C_G(x)/C^{^{\wedge}}_G(x)\) is isomorphic to a subgroup of \(M(G)\); in particular if \(G\) has an element \(x\) such that \(C^{^{\wedge}}_G(x)\lneqq C_G(x)\) then \(M(G)\neq 1\). Some other related results are proved.
scientific article; zbMATH DE number 5655345