# Equivalence of definitions of conjugacy functor whose normalizer generates whole group with p'-core

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term conjugacy functor whose normalizer generates whole group with p'-core

View a complete list of pages giving proofs of equivalence of definitions

## Statement

Suppose is a group, is a prime number, and is a conjugacy functor for . The following conditions are equivalent, where is any -Sylow subgroup of .

- The image of in the quotient is a normal subgroup of .

## Facts used

## Proof

### (2) implies (1)

**Given**: A prime , a finite group such that if , then for every -Sylow subgroup of , the image of is normal in .

**To prove**: For one (and hence every) -Sylow subgroup of , .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | Let , so , and be the quotient map, with . | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
| |||

2 | Let . In other words, . | ||||

3 | is normal in . | Fact (1) | is normal in . | Steps (5), (6) | fact-step combination direct. |

4 | is -Sylow in . | Step (2) | [SHOW MORE] | ||

5 | . | Fact (2) | Steps (3), (4) | Fact-step-combination direct. | |

6 | Steps (2), (5) | [SHOW MORE] |