On generalized Dedekind groups and Tarski super monsters (Q1977538)

A group \(G\) is called a \(J\)-group if it satisfies the following condition: for each element \(x\in G\) either the subgroup \(\langle x\rangle\) is normal in \(G\), or the subgroup \(\langle x,x^g\rangle\) is normal in \(G\) for all elements \(g\in G\setminus N_G(\langle x\rangle)\). \(J\)-groups are a generalization of the Dedekind groups. The first step is the study of finite \(J\)-groups. Theorem 1. If \(G\) is a finite \(J\)-group, then \(G\) is soluble. Theorem 2. Let \(G\) be a finite nilpotent group. Then \(G\) is a \(J\)-group if and only if all the Sylow subgroups of \(G\) are \(J\)-groups and all but at most one are Dedekind groups. Theorem 5. Let \(G\) be a finite \(J\)-group, \(Z\) be the upper hypercenter of \(G\). Then \(G/Z\) is a Frobenius group with an elementary Abelian kernel \(P\) of order \(p^i\), \(i\leq 2\), for some prime \(p\), and with a cyclic complement \(D\). If \(|P|=p^2\), then \(D\) is of odd order and \(|D|\) divides \(p+1\). Theorem 6. Let \(G\) be a finite group with identity center. Then \(G\) is a \(J\)-group if and only if \(G\) is a Frobenius group with an elementary Abelian kernel \(P\) of order \(p\) or \(p^2\) and a cyclic complement \(D\), which is of odd order dividing \(p+1\) if \(|P|=p^2\). Section III of this paper contains some properties of finite primary \(J\)-groups. Section IV of this paper contains some properties of infinite soluble \(J\)-groups. The basis is Theorem 24. Let \(G\) be a finitely generated infinite non-Abelian group. Then \(G\) is a \(J\)-group if and only if \([G,G]\) has prime order. Corollary 25. Let \(G\) be a locally soluble \(J\)-group. Then (i) \(G\) is soluble; (ii) \(\text{dl}(G)\leq 3\); (iii) if \(G\) is not periodic then either \(G\) is Abelian, or \([G,G]\) has prime order. Corollary 29. Let \(G\) be a locally nilpotent \(J\)-group. Then (i) \(G\) is nilpotent; (ii) \(\text{cl}(G)\leq 3\). Theorem 30. Let \(G\) be an infinite locally finite \(J\)-group. Then (i) \(G\) is soluble and \(\text{dl}(G)\leq 3\); (ii) either \(G=Z_3(G)\) or \(G/Z_3(G)\) is a finite Frobenius group of the special type described in Theorem 5. Section V of this paper contains some properties of infinite non-soluble \(J\)-groups.