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2003•Illinois Journal of MathematicsOpen access

Periodic groups with nearly modular subgroup lattice

Maria De Falco, Carmela Musella, Yaroslav Sysak, Francesco de Giovanni

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Abstract

A theorem of B.H. Neumann states that each subgroup of a group $G$ has finite index in a normal subgroup of $G$ if and only if the commutator subgroup $G'$ of $G$ is finite, i.e., $G$ is finite-by-abelian. As a group lattice version of this theorem for a periodic group $G$, it is proved that each subgroup of $G$ has finite index in a modular subgroup of $G$ if and only if $G$ is an extension of a finite group by a group with modular subgroup lattice.

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A theorem of B.H. Neumann states that each subgroup of a group $G$ has finite index in a normal subgroup of $G$ if and only if the commutator subgroup $G'$ of $G$ is finite, i.e., $G$ is finite-by-abelian. As a group lattice version of this theorem for a periodic group $G$, it is proved that each subgroup of $G$ has finite index in a modular subgroup of $G$ if and only if $G$ is an extension of a finite group by a group with modular subgroup lattice.

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Available abstract

A theorem of B.H. Neumann states that each subgroup of a group $G$ has finite index in a normal subgroup of $G$ if and only if the commutator subgroup $G'$ of $G$ is finite, i.e., $G$ is finite-by-abelian. As a group lattice version of this theorem for a periodic group $G$, it is proved that each subgroup of $G$ has finite index in a modular subgroup of $G$ if and only if $G$ is an extension of a finite group by a group with modular subgroup lattice.

Key concepts: Commutator subgroup, Mathematics, Characteristic subgroup, Index of a subgroup, Fitting subgroup, Normal subgroup, Maximal subgroup, Omega and agemo subgroup

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