Strongly 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 an odd prime number. We say that $G$ is strongly $p$-solvable if it satisfies both the following conditions:

• $G$ is a p-solvable group.
• Either $p\geq 5$ or $p=3$ and no subquotient of $G$ is isomorphic to special linear group:SL(2,3).

Note that for $p=2$, there is no notion of strong solvability.

Relation with other properties

Weaker properties

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

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 234, Chapter 6 (Solvable and pi-solvable groups), Section 6.5 (p-stability in p-solvable groups), ^{More info}