On subnormality criteria for subgroups in finite groups

Abstract

Let H be a subgroup of a finite group G

Nilpotence relations in products of groups

“…Consider G to be the symmetric group on a set of 8 elements S 8 . As we already mentioned Fumagalli, in [24], observed that considering and calling H := h and A := a, b , we obtain that G = A, H . With a little calculation we proved that A and H are N 3 -connected subgroups, i.e.…”

“…In fact: consider a non nilpotent group G. If we put G = A = B, then A and B are trivially cosubnormal, but they are not N -connected. For the vice versa, the authors construct an example in [11] or alternatively we propose the following example in [24] Called H := h and A := a, b , it is possible to show that G = A, H and A and H are N 3 -connected subgroups, but for example H is not subnormal in G.…”

“…For instance, let G be the symmetric group of degree 4. Choose A := (12) B := (123) C := (12)(34), (13)(24)…”

On $$\sigma $$-subnormality criteria in finite $$\sigma $$-soluble groups

“…Consequently X, X g is a σ i -group and then X is σ-subnormal in X, X g for every g ∈ G by Lemma 1 (6). Applying Theorem A, X is σ-subnormal in G, a contradiction.…”