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maximal subgroup
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(Definition)
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Let $G$ be a group.
A subgroup $H$ of $G$ is said to be a maximal subgroup of $G$ if $H\neq G$ and there is no subgroup $K$ of $G$ such that $H<K<G$ . Note that a maximal subgroup of $G$ is not maximal among all subgroups of $G$ , but only among all proper subgroups of $G$ . For this reason, maximal subgroups are sometimes called maximal proper subgroups.
Similarly, a normal subgroup $N$ of $G$ is said to be a maximal normal subgroup (or maximal proper normal subgroup) of $G$ if $N\neq G$ and there is no normal subgroup $K$ of $G$ such that $N<K<G$ . We have the following theorem:
Theorem A normal subgroup $N$ of a group $G$ is a maximal normal subgroup if and only if the quotient $G/N$ is simple.
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"maximal subgroup" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: maximal element
| Other names: |
maximal proper subgroup |
| Also defines: |
maximal, maximal normal subgroup, maximal proper normal subgroup, simplicity of quotient group |
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Cross-references: normal subgroup, proper subgroups, subgroup, group
There are 23 references to this entry.
This is version 12 of maximal subgroup, born on 2002-02-19, modified 2009-06-03.
Object id is 2198, canonical name is Maximal.
Accessed 16615 times total.
Classification:
| AMS MSC: | 20E28 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Maximal subgroups) |
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Pending Errata and Addenda
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