Definition:Maximal Subgroup

Definition

Let $G$ be a group.

Let $M \le G$ be a proper subgroup of $G$.

Then $M$ is a maximal subgroup of $G$ if and only if:

For every subgroup $H$ of $G$, $M \subseteq H \subseteq G$ means $M = H$ or $H = G$.

That is, if and only if there is no subgroup of $G$, except $M$ and $G$ itself, which contains $M$.

Also see

• Definition:Maximal Set
• Definition:Minimal Set

• Definition:Minimal Subgroup

• Results about maximal subgroups can be found here.

Sources

• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$
• 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation