On the capability of finite abelian pairs of groups (Q2833993)

A group is called \textit{capable} if it is isomorphic to the group of inner automorphisms of another group. A pair \((G,N)\) with a normal subgroup \(N\) of a group \(G\) is called \textit{capable} if it admits a relative central extension of the pair \((G,N)\), that is, a group homomorphism \(\partial :M\rightarrow G\), together with an action of \(G\) on \(M\), with: (i) \( \partial (M)=N\); (ii) \(\partial (^{g}m)=g(\partial m)g^{-1}\) for every \(g\in G\), \(m\in M\); (iii) \(^{(\partial m)}m^{\prime }=mm^{\prime }m^{-1}\) for every \(m,m^{\prime }\in M\), and (iv) \(\ker (\partial )\subseteq Z_{G}(M)\), such that \(\ker (\partial )=Z_{G}(M)\). This way, a group \(K\) is capable precisely if the pair \((K,K)\) is capable.NEWLINENEWLINE\textit{R. Baer} [Trans. Am. Math. Soc. 44, 387--412 (1938; Zbl 0020.00803)] determined all capable groups which are direct sums of cyclic groups, therefore including all finite abelian groups. Recently, \textit{Z. Šunić} [Arch. Math. 93, No. 1, 23--28 (2009; Zbl 1188.20063)] provided a different characterization of capable finite abelian groups by considering a condition on the lattice of subgroups.NEWLINENEWLINEIn [Bull. Malays. Math. Sci. Soc. (2) 35, No. 1, 205--213 (2012; Zbl 1239.20035)], \textit{A. Pourmirzaei} et al. determined all capable pairs of finitely generated abelian groups.NEWLINENEWLINEIn this paper, a different characterization of capable pairs of finite abelian groups is given, in a similar way to Šunić's above mentioned result [loc. cit.].