Finite solvable group: Difference between revisions
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==Definition== | ==Definition== | ||
A [[finite group]] is termed a '''finite solvable group''' if it satisfies the following equivalent conditions: | A [[defining ingredient::finite group]] is termed a '''finite solvable group''' if it satisfies the following equivalent conditions: | ||
# It is a [[defining ingredient::solvable group]] | |||
# It is a [[defining ingredient::polycyclic group]] | |||
# It has [[Sylow complement]]s for all prime divisors of the order of the group | |||
# It has [[Hall subgroup]]s of all possible orders | |||
# All its composition factors (i.e., the quotient groups for any [[defining ingredient::composition series]] for the group) are cyclic groups of prime order. | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 00:58, 15 November 2008
This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
View other properties of finite groups OR View all group properties
Definition
A finite group is termed a finite solvable group if it satisfies the following equivalent conditions:
- It is a solvable group
- It is a polycyclic group
- It has Sylow complements for all prime divisors of the order of the group
- It has Hall subgroups of all possible orders
- All its composition factors (i.e., the quotient groups for any composition series for the group) are cyclic groups of prime order.