Strongly 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
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Contents
Definition
Let be a finite group and be an odd prime number. We say that is strongly -solvable if it satisfies both the following conditions:
- is a p-solvable group.
- Either or and no subquotient of is isomorphic to special linear group:SL(2,3).
Note that for , 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}