P-solvable group

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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

Definition

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