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This note provides an affirmative answer to Problem 2.6 of Praeger and Schneider ['Group factorisations, uniform automorphisms, and permutation groups of simple diagonal type', Israel J. Math. 228 (2) (2018), 1001–1023]. We will build groups $G$ (abelian, nonabelian and simple) for which there are two automorphisms $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}$ of $G$ such that the map $$\begin{eqnarray}T=T_{\unicode[STIX]{x1D6FC}}\times T_{\unicode[STIX]{x1D6FD}}:G\longrightarrow G\times G,\quad g\mapsto (g^{-1}g^{\unicode[STIX]{x1D6FC}},g^{-1}g^{\,\unicode[STIX]{x1D6FD}})\end{eqnarray}$$ is surjective. [ABSTRACT FROM AUTHOR]

A finite group is called a CH-group if for every x, y ϵ G\Z(G), xy = yx implies that ∥CG(x)∥ = ∥CG(y)∥. Applying results of Schmidt ['Zentralisatorverbände endlicher Gruppen', Rend. Sem. Mat. Univ. Padova 44 (1970), 97-131] and Rebmann ['F-Gruppen', Arch. Math. 22 (1971), 225-230] concerning CA-groups and F-groups, the structure of CH-groups is determined, up to that of CH-groups of prime-power order. Upper bounds are found for the derived length of nilpotent and solvable CH-groups. [ABSTRACT FROM AUTHOR]