Chordal Enhanced Power Graphs of Finite Simple Groups and Induced 4-Cycles: On Kourovka Notebook Problem 21.23

For a finite group \(G\), the enhanced power graph \(E(G)\) has vertex set \(G\), two distinct elements being adjacent when they generate a cyclic subgroup. Bubboloni, Fumagalli and Praeger asked (Kourovka Notebook, Problem 21.23) for which finite nonabelian simple groups \(E(G)\) has no induced cycle of a given length \(n\ge 4\), and for which \(E(G)\) is chordal. By their results, \(E(G)\) is chordal for \(\mathrm{PSL}_2(q)\) (\(q\ge 4\)), the Suzuki groups, \(\mathrm{PSL}_3(4)\) and \(A_7\). We show that \(E(G)\) has an induced cycle of length at least \(4\) for \(M_{11}\), \(J_1\), the Tits group, \(\mathrm{PSL}_3(q)\) with \(q\) odd, \(\mathrm{PSU}_3(q)\) with \(q\ge 3\), and the Ree groups \({}^2G_2(q)\) with \(q\ge 27\); for the infinite families the cycles are given by direct constructions. Combined with the inclusion, proved by Bubboloni, Fumagalli and Praeger in an unrefereed preprint by a case analysis over the classification of finite simple groups, of all simple groups without induced 4-cycles in an explicit list, this gives: \(E(G)\) is chordal if and only if \(G\) is one of the groups of the first list. For \(n=4\) we show that \(E(G)\) has no induced \(4\)-cycle for the Tits group and, using the published lists of maximal subgroups of Bray, Holt and Roney-Dougal, for \(\mathrm{PSL}_3(p)\) and \(\mathrm{PSU}_3(p)\) when \(p\) is a prime with \(p-1\), respectively \(p+1\), equal to \(2^a\) or \(3\cdot 2^a\), while it has one for all other \(\mathrm{PSU}_3(q)\) and for \({}^2G_2(q)\) with \(q\ge 27\); under the same two assumptions this determines the set for \(n=4\). For \(n\ge 5\) the question remains open; we determine the least length of an induced cycle of length at least 4 for some small groups. Several steps are computer-assisted, and all certificates are included. This is an unrefereed note.