On the structure of groups whose non-normal subgroups are core-free (Q2334584)

If \(H\) is a subgroup of a group \(G\), then \(H_G\) denotes the largest normal subgroup of \(G\) contained in \(H\). This paper studies groups \(G\), which for every subgroup \(H\) of \(G\) the normal subgroup \(H_G\) is \(\langle 1\rangle\) or \(H\). For such \(G\) if \(G\) is infinite and if \(M(G)\), the intersection of all the non-trivial normal subgroups of \(G\), is \(\langle 1\rangle\), then (Theorem 1.1) \(G\) is a Dedekind group. The remainder of the paper describes in great detail the infinite locally soluble such groups with \(M(G) \neq\langle 1\rangle\). The conclusions are different (and far to long to state here) according to whether \(G\) is periodic (Theorem 1.4) or not periodic (Theorem 1.2). Finally, Theorem 1.5 describes in detail the case where \(G\) is finite and soluble.