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 be a finite group and be a prime number. We say that is a p-solvable group if it satisfies the following equivalent conditions:
- has a subnormal series where all the quotients are either -groups or have orders relatively prime to .
- is a pi-separable group for . In other words, has a p-series.
- All the composition factors of that are non-abelian do not have dividing their order.
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 -series that minimizes this number.