On Intersection of Abelian and Minimal Nonabelian Subgroups in Finite Groups

Zenkov, V. I.
Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 3.

Abstract:
Let \(G\) be a finite group with subgroups \(A\) and \(B\). Denote by \(M=M_G(A,B)\) (respectively, \(m=m_G(A,B)\)) the set of all minimal by inclusion (respectively, by order) intersections of the form \(A\cap B^g\), where \(g\in G\). Put \(\min_G(A,B)=\langle m\rangle\) and \(\mbox{\rm Min}_G(A,B)=\langle M\rangle\). In 1994 we proved that if \(A\) and \(B\) are abelian subgroups, then \(\mbox{\rm Min}_G(A,B)\le F(G)\). In the present paper, we give other proof of this result. Futhermore, we construct a finite group \(G\) such that it contan an abelian subgroup \(A\), a minimal non-abelian subgroup \(B\) and elements \(g_1\) and \(g_2\) with \(A\cap B^{g_1}\le F(G)\), \(A\cap B^{g_2}\not\le F(G)\), \(|A\cap B^{g_1}|=|A\cap B^{g_2}|\) and \(A\cap B^{g_1}\), \( A\cap B^{g_2}\in\min_G(A,B)\). We provide an example of a group \(G\) such that \(g_1, g_2\in G\) \(A\cap B^{g_1}\), \( A\cap B^{g_2}\in\mbox{\rm Min}_G(A,B)\), \(A\cap B^{g_1}\le F(G)\), and \(A\cap B^{g_2}\not\le F(G)\). Moreover, we show that there exists a group \(G\) with nilpotent subgroups \(A\) and \(B\) such that \(m\subset M\) and \(\min_G(A,B) < \mbox{\rm Min}_G(A,B)\).

Keywords: finite group, abelian subgroup, intersection of subgroups

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