P-solvable implies p-constrained
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., p-solvable group) must also satisfy the second group property (i.e., p-constrained group)
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Statement with symbols
Suppose is a finite group and is a prime number. Suppose further that is -solvable. Then, if is a -Sylow subgroup, we have:
In other words, is -constrained.
- p-constrained not implies p-solvable
- The converse is partially true: p-constrained implies not simple non-abelian
- Equivalence of definitions of Sylow subgroup of normal subgroup: This states that the intersection of a Sylow subgroup and a normal subgroup is a Sylow subgroup of the normal subgroup.
- Sylow satisfies image condition
- Pi-separable and pi'-core-free implies pi-core is self-centralizing
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Given: A finite group that is -solvable for some prime . is a -Sylow subgroup.
To prove: Let . Then, , where is the centralizer of in .
Proof: Let be the natural quotient map. Note that .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||is -core-free||[SHOW MORE]|
|2||is a -Sylow subgroup of||Fact (1)||is -Sylow in , and .||[SHOW MORE]|
|3||, or equivalently .||Fact (2)||Step (2)||[SHOW MORE]|
|4||is self-centralizing, i.e.,||Fact (3)||is -solvable.||Steps (1), (3)||[SHOW MORE]|
|5||This follows from the definition of homomorphism: if an element centralizes , its image centralizes the image of .|
|6||Steps (4), (5)||Step-combination direct|
|7||Steps (3), (6)||Step-combination direct|