On minimal intersections of pairs of primary subgroups in \(Aut(L_n(2))\)

V. Zenkov

We consider finite groups only. In [1], the following theorem about abelian subgroups was proved: in every finite group \(G\), for every abelian subgroups \(A\) and \(B\), the minimal by inclusion intersections of the form \(A\cap B^g\), where \(g\in G\), lie in the Fitting subgroups \(F(G)\) of \(G\). If we relax the condition of being abelian for at least one of the subgroups, which we may even take minimal nonabelian, then the conclusion fails. For example, in the group \(G=S_4\), for a subgroup \(A\cong D_8\) of \(G\) and a subgroup \(B\) of \(A\) of order four outside of \(O_2(G)\), we have \(B=A\cap B\not\le O_2(G)=F(G)\). Furthermore, in this group, the set of minimal by inclusion intersections of the form \(A\cap B^g\), where \(g\in G\), generates the subgroup \(Min(A,B)\) equal to \(A\). At the same time, the set of intersections of the form \(A\cap B^g\), where \(g\in G\), of minimal order generates the subgroup \(min(A,B)\) equal to \(O_2(G)=F(G)\) and distinct from \(A=Min(A,B)\). Therefore, \(Min(A,B)>min(A,B)\) in this particular example. In [2], it is proved that, in the group \(Aut(A_8)\cong S_8\), every pair of Sylow 2-subgroups intersects nontrivially. In particular, for a Sylow 2-subgroup \(S\) of such a group, we have \(min(S,S)\ne 1\). Observe that \(Aut(A_8)\cong L_4(2)\) and, for the series of groups \(L_n(2)\) with \(n>2\), it is proved in [3, Theorem B2] that every pair of Sylow 2-subgroups of such a group intersects nontrivially, or, equivalently, that \(Min(S,S)\ne 1\) for a Sylow 2-subgroup \(S\) of every group \(G\) in this series. In particular, \(min(S,S)\ne 1\) for every group \(G\) in this series. The definition of \(Min(A,B)\) implies that, in a group \(G\) with a Sylow subgroup \(S\), a nontrivial subgroup \(Min(S,S)\) intersects nontrivially with every subgroup conjugate to \(S\) in \(G\). In general, a nontrivial subgroup \(min(S,S)\) may intersect trivially with some subgroup conjugate to \(S\) in \(G\). This happens, for example, in the groups \(G\) that is the extension of \(L_3(4)\) by the subgroup generated by the graph automorphism of order 2. Nevertheless, in \(Aut(L_n(2)), n>2\), with a Sylow subgroup \(S\), the subgroup \(min(S,S)\) intersects nontrivially with every subgroup conjugate to \(S\) in \(G\), and, for primary subgroups that are not overgroups of \(min(S,S)\), an analog of the above-mentioned theorem about abelian subgroups of finite groups holds. More precisely, the following theorem holds.

Theorem. Let \(G\) be a finite group isomorphic to \(Aut(L_n(2)), n>2\), let \(S\) be a Sylow \(2\)-subgroup of \(G\), and let \(A\) and \(B\) be primary subgroups of \(G\). Then the following conditions are equivalent:
\(\ \ 1)\ \ \) \(A\cap B^g\ne 1\) for every \(g\in G\);
\(\ \ 2)\ \ \) either \(G\cong Aut(L_{2m+1}(2))\), where \(m>0\) and to within conjugacy \((A,B)=(S,S)\) and \(min(S,S)=S\), or \(G\cong Aut(L_{2m}(2))\), where \(m>1\) and, for a minimal non-\(2\)-closed parabolic subgroup \(P\) of \(G'\) that contains \(S\cap G'\) and for an involution \(g\) that induces a graph automorphism on \(G'\) and a symmetry on the Dynkin diagram for \(G'\) such that \(g\) normalizes \(P\), to within conjugacy the pair \((A,B)\) lies in the set \[\{(S,S),(min(S,S),S), (S, min(S,S)),(min(S,S),min(S,S))\},\] where \(min(S,S)=O_2(N_G(P))\).

Remark. If \(G=Aut(A_8)\) then \(min(S,S)=O_2(C_G(z))\), where \(z\) is the involution in \(Z(S)\). Moreover, \(O_2(C_G(z))\) is generated by an involution \(g\) that induces a graph automorphism on \(A_8\cong L_4(2)\) and by the closure in \(S\) of an involution \(j=z^g\) such that \(|S\cap G':C_{S\cap G'}(j)|=4\), i. e., \(O_2(C_G(z))=\langle g, j^S\rangle\).

Acknowledgement. The work is supported by RFBR (project 13-01-00469), by the Program of the Division of Mathematical Sciences of RAS (project 12-T-1-1003), and by the Joint Research Program of UB RAS with SB RAS (project 12-C-1-10018) and with NAS of Belarus (project 12-C-1-1009).

References

1. V. I. Zenkov, Intersection of Abelian subgroups in finite groups, Math. Notes, 56, N2 (1994), 869–871.
2. V. I. Zenkov, V. D. Mazurov, On the intersection of Sylow subgroups in finite groups, Algebra Logic, 35, N4 (1996), 236–240.
3. V. I. Zenkov, Intersections of nilpotent subgroups in finite groups, Fund. Prikl. Mat., 2, N1 (1996), 1–92. (Russian)

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