Schur multipliers and the Lazard correspondence.
Let \(G\) be a finite \(p\)-group of nilpotency class less than \(p\), and let \(L\) be a Lie ring of \(p\)-power order of nilpotency class less than \(p\). The authors study properties behaving regularly under the Lazard correspondence \(G\mapsto L=\mathrm{Lie}(G)\mapsto\mathrm{Grp}(L)=G\). Throughout the paper it is assumed that the nilpotency class is less than \(p-1\). The main results are as follows. Schur multipliers of \(G\) and \(\mathrm{Lie}(G)\) are isomorphic as abelian groups. Isomorphism classes of Schur-covers of \(G\) and \(\mathrm{Grp}(L)\) correspond to each other under Lazard correspondence. A group (Lie ring) is capable if it is a central factor of some group (Lie ring). The epicenter of a group (Lie ring) is the intersection of all normal subgroups (ideals) with capable factor. The authors also show that epicenters of \(G\) and \(\mathrm{Lie}(G)\) are isomorphic as abelian groups, and capability is invariant under Lazard correspondence. These results are applied to groups of order \(p^5\) and class 3. The paper is very well written.
scientific article; zbMATH DE number 6109273