P-solvable group

From Groupprops
Revision as of 22:33, 8 August 2012 by Vipul (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
View other prime-parametrized group properties | View other group properties


Let G be a finite group and p be a prime number. We say that G is a p-solvable group if it satisfies the following equivalent conditions:

Note that if p does not divide the order of G, G is p-solvable. Further, a finite group is a finite solvable group if it is p-solvable for every prime p dividing the order of G.

There is a notion of p-length to measure the length of a p-solvable group; briefly, it measures the number of the successive quotient groups that are p-groups in any p-series that minimizes this number.

Relation with other properties

Stronger properties

Weaker properties