On finite groups with some conditions on subsets. (Q998803)

For any positive integer \(n\), the author in the paper under review defines a class of groups denoted by \(W^*(n)\) to be the class of groups \(G\) such that for every subset \(X\) of \(G\) of cardinality \(n+1\) there exist a positive integer \(k\) and a subset \(X_0\subseteq X\) with \(2\leq|X_0|\leq n+1\) and a function \(f\colon\{0,1,\dots,n\}\to X_0\) with \(f(0)\neq f(1)\) and non-zero integers \(t_0,t_1,\dots t_k\) such that \([x_0^{t_0},x_1^{t_1},\dots,x_k^{t_k}]=1\) where \(x_i:=f(i)\), \(i=0,\dots,k\) and \(x_j \in H\) whenever \(x_j^{t_j}\in H\) for some subgroup \(H\neq\langle x_j^{t_j}\rangle\) of \(G\). If the integer \(k\) is fixed for every subset \(X\), the class of all such groups \(G\) is denoted by \(W^*_k(n)\); and if one always has \(t_i=1\) for all \(i\in\{0,\dots,k\}\) and \(x_0=x\), \(x_i=y\) for all \(i\in\{1,\dots,k\}\), then the class of all such groups \(G\) is denoted by \(\mathcal E_k(n)\). The author proves that: 1) A finite semi-simple group has the property \(W^*_k(5)\) for some \(k\) if and only if \(G\cong A_5\) or \(S_5\). 2) A finite non-nilpotent group has the property \(W^*_k(3)\) for some \(k\) if and only if \(G/Z^*(G)\cong S_3\) where \(Z^*(G)\) is the hypercenter of \(G\). 3) A finite semi-simple group has the property \(\mathcal E_k(16)\) for some \(k\) if and only if \(G\cong A_5\). Here \(A_n\) and \(S_n\) denote the alternating group and the symmetric group of degree \(n\), respectively, and by a semi-simple group it is meant a group with no non-trivial normal Abelian subgroup. Some similar classes of groups, defined combinatorially, have been studied [in Houston J. Math. 27, No. 3, 511-522 (2001; Zbl 0999.20028), Bull. Aust. Math. Soc. 62, No. 1, 141-148 (2000; Zbl 0964.20019) and J. Algebra 283, No. 2, 431-446 (2005; Zbl 1116.20030)] by the reviewer.