P-solvable group

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:


 * $$G$$ has a subnormal series where all the quotients are either $$p$$-groups or have orders relatively prime to $$p$$.
 * $$G$$ is a defining ingredient::pi-separable group for $$\pi = \{ p \}$$. In other words, $$G$$ has a p-series.
 * All the defining ingredient::composition factors of $$G$$ that are non-abelian do not have $$p$$ dividing their order.

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.

Stronger properties

 * Weaker than::Strongly p-solvable group

Weaker properties

 * Stronger than::p-constrained group: