Frattini subgroup: Difference between revisions
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In [[group theory]], the '''Frattini subgroup''' is the intersection of all maximal [[subgroup]]s of a group. | In [[group theory]], the '''Frattini subgroup''' is the intersection of all maximal [[subgroup]]s of a group. | ||
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:<math>\Phi(G) = G \cap \bigcap_M M \,</math> | :<math>\Phi(G) = G \cap \bigcap_M M \,</math> | ||
where ''M'' runs over all maximal | where ''M'' runs over all [[maximal subgroup]]s of G. If ''G'' has no maximal subgroups then <math>\Phi(G) = G</math>. | ||
The Frattini is a [[subgroup]], which is [[normal subgroup|normal]] and indeed [[characteristic subgroup|characteristic]]. | The Frattini is a [[subgroup]], which is [[normal subgroup|normal]] and indeed [[characteristic subgroup|characteristic]]. | ||
==References== | ==References== | ||
* {{cite book | author=Marshall Hall jr | title=The theory of groups | publisher=Macmillan | location=New York | year=1959 | pages=156 }} | * {{cite book | author=Marshall Hall jr | title=The theory of groups | publisher=Macmillan | location=New York | year=1959 | pages=156 }}[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:00, 18 August 2024
In group theory, the Frattini subgroup is the intersection of all maximal subgroups of a group.
Formally,
where M runs over all maximal subgroups of G. If G has no maximal subgroups then .
The Frattini is a subgroup, which is normal and indeed characteristic.
References
- Marshall Hall jr (1959). The theory of groups. New York: Macmillan, 156.