Difference between revisions of "Finite solvable group"

From Groupprops
Jump to: navigation, search
m (1 revision)
Line 3: Line 3:
 
==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 [[solvable group]]
+
# It is a [[defining ingredient::solvable group]]
* It is a [[polycyclic 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 [[Sylow complement]]s for all prime divisors of the order of the group
* It has [[Hall subgroup]]s of all possible orders
+
# 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:

  1. It is a solvable group
  2. It is a polycyclic group
  3. It has Sylow complements for all prime divisors of the order of the group
  4. It has Hall subgroups of all possible orders
  5. All its composition factors (i.e., the quotient groups for any composition series for the group) are cyclic groups of prime order.

Relation with other properties

Stronger properties