On maximal subgroups of minimax groups
On maximal subgroups of minimax groups
It is known, that if \(G\) is a finitely generated soluble group such that \(G/\Phi(G)\) is nilpotent then \(G\) is nilpotent too, where \(\Phi(G)\) is the Frattini subgroup of the group \(G\). J. C. Lennox proved that if \(G\) is a finitely generated soluble group such that \(G/\Phi(G)\) is finite-by- nilpotent then \(G\) is finite-by-nilpotent. The main result of this paper is Theorem. Let \(G\) be a soluble residually finite minimax group. Then \(G\) is finite-by-nilpotent if and only if it has finitely many maximal subgroups which are not normal. Corollary. Let \(G\) be a soluble residually finite minimax group such that the factor- group \(G/\Phi(G)\) is finite-by-nilpotent. Then \(G\) is finite-by- nilpotent.
Frattini subgroup, Generalizations of solvable and nilpotent groups, Solvable groups, supersolvable groups, Chains and lattices of subgroups, subnormal subgroups, finitely generated soluble groups, Subgroup theorems; subgroup growth, soluble residually finite minimax groups, finite-by-nilpotent groups, Maximal subgroups, Nilpotent groups, maximal subgroups, Residual properties and generalizations; residually finite groups
Frattini subgroup, Generalizations of solvable and nilpotent groups, Solvable groups, supersolvable groups, Chains and lattices of subgroups, subnormal subgroups, finitely generated soluble groups, Subgroup theorems; subgroup growth, soluble residually finite minimax groups, finite-by-nilpotent groups, Maximal subgroups, Nilpotent groups, maximal subgroups, Residual properties and generalizations; residually finite groups
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