Definition:Minimal Subgroup
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Definition
Let $G$ be a group.
Let $M \le G$ be a non-trivial subgroup of $G$.
Then $M$ is a minimal subgroup of $G$ if and only if:
- For every subgroup $H$ of $G$, $H \subseteq M$ means $H = M$ or $H = \set e$.
That is, if and only if there is no subgroup of $M$, except $M$ and $\set e$ itself, which is a subset of $M$.
Also see
- Results about minimal subgroups can be found here.
Sources
- 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation