On the periodic groups saturated with finite simple groups of Lie type \(B_3\). (English. Russian original) Zbl 1481.20136

Lytkina, D. V.; Mazurov, V. D.

Sib. Math. J. 61, No. 3, 499-503 (2020); translation from Sib. Mat. Zh. 61, No. 3, 634-640 (2020).

Summary: Let \(\mathfrak{M}\) be a set of finite groups. Given a group \(G\), denote by \(\mathfrak{M}(G)\) the set of all subgroups of \(G\) isomorphic to the elements of \(\mathfrak{M} \). A group \(G\) is said to be saturated with groups from \(\mathfrak{M} \) (saturated with \(\mathfrak{M} \), for brevity) if each finite subgroup of \(G\) lies in an element of \(\mathfrak{M}(G)\). We prove that a periodic group \(G\) saturated with \(\mathfrak{M}=\left\{O_7(q)\mid{q}\equiv\pm3 \pmod 8)\right\}\) is isomorphic to \(O_7(F)\) for some locally finite field \(F\) of odd characteristic.

MSC:

20F50	Periodic groups; locally finite groups
20E32	Simple groups
20D06	Simple groups: alternating groups and groups of Lie type

Keywords:

periodic group; group of Lie type; orthogonal group; group saturated with set of groups

Software:

GAP

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