P-solvable group

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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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 pi-separable group for $\pi =\{p\}$. In other words, $G$ has a p-series.
• All the 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.

Relation with other properties

Stronger properties

• Strongly p-solvable group

Weaker properties

• p-constrained group: For proof of the implication, refer p-solvable implies p-constrained and for proof of its strictness (i.e. the reverse implication being false) refer p-constrained not implies p-solvable.