# P-solvable implies p-constrained

From Groupprops

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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## Contents

## Statement

### Verbal statement

Any p-solvable group is a p-constrained group.

### 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.

## Related facts

### Converse

- p-constrained not implies p-solvable
- The converse is
*partially*true: p-constrained implies not simple non-abelian

## Facts used

- 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

## Proof

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

**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 |