Split metacyclic \(p\)-groups that are \(A\)-\(E\) groups (Q1306286)

For a given group \(G\), there are 3 near-rings generated by suitable sets of endomorphisms which are of particular interest. In order of increasing size they are: \(I(G)\) generated by all the inner automorphisms, \(A(G)\) generated by all the automorphisms and \(E(G)\) generated by all the endomorphisms. A question going back at least as far as Fitting asks when are two of these the same? The author has made major contributions to this study [Commun. Algebra 23, No.~12, 4563-4585 (1995; Zbl 0848.16037) and Monatsh. Math. 121, No.~3, 275-290 (1996; Zbl 0927.16038)] and here he deals with the case when \(G\) is a split metacyclic \(p\)-group that is an \(A\)-\(E\) group, so \(A(G)=E(G)\). By using presentations due to \textit{B. W. King} [Bull. Aust. Math. Soc. 8, 103-131 (1973; Zbl 0245.20016)] and some very useful and intricate analysis, he is able to determine all the automorphisms and endomorphisms of these groups, results which are of independent interest. He then goes on to show that the groups concerned are \(A\)-\(E\)-groups if and only if a certain set of simultaneous linear equations have a solution over the ring of integers modulo \(p^m\). As a corollary, he shows that there are no split metacyclic 2-groups that are \(A\)-\(E\)-groups. However if \(p\) is an odd prime then all split metacyclic \(p\)-groups of nilpotency class at most \(p-1\) are \(A\)-\(E\)-groups. For higher nilpotency class examples are given to show that both cases (\(A\)-\(E\) and not \(A\)-\(E\)) are possible.