On infinitely generated groups whose proper subgroups are solvable

A Python Code for Generating All Proper Subgroups of Dihedral Group

“…A proper subgroup is a subgroup that is distinct from the group itself. Many author previously have consider such variety of proper subgroup as can be seen in [2,3,4,5,6,7].…”

On Minimal Non-(residually Nilpotent) Locally Graded Groups

“…Finally, Theorem 1.1 is used to obtain a short proof and a generalization of [2, Theorem 1.1(a)](see Theorem 1.6). Of course if in the hypothesis of Theorem 1.6, F G is residually finite instead of being residually nilpotent for every finite subgroup F , then G is solvable by [2,Corollary 1.5].…”

“…(Note also that [2, Lemma 2.8] holds when G is not perfect). (See [2] for some properties of a ( * )-triple). To see another property of a ( * )-triple, let G be a Fitting p-group and (w, V, L) be a ( * )-triple in G. Let A be a normal abelian subgroup of G. Then A/(A∩L) is locally cyclic-by-finite.…”

“…Theorem 1.2 is a partial generalization of [2,Theorem 1.2]. Let G be a perfect totally imprimitive p-subgroup of F Sym(Ω), where |Ω| = ∞ such that every subgroup of G having an infinite orbit is transitive.…”

“…Suppose that G has a ( * )-triple (w, V, L) such that L satisfies the normalizer condition. (Note that G always contains ( * )-triples either by Proposition A and [7,Theorem] or by [2,Lemma 2.8]). Now it follows from by [7,Lemma] G is finite which impossible since G is transitive on an infinite set.…”