# Congruence condition on index of subgroup containing Sylow-normalizer

From Groupprops

## Statement

Suppose is a finite group, is a prime number, is a -Sylow subgroup, and is a subgroup of such that . In other words, contains a -Sylow normalizer (?). Then, the index of in is congruent to modulo .

## Facts used

- Congruence condition on Sylow numbers
- Sylow satisfies intermediate subgroup condition: Any -Sylow subgroup of a group is also a -Sylow subgroup in any intermediate subgroup.
- Index is multiplicative

## Proof

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**Given**: A finite group , a prime , a -Sylow subgroup , a subgroup of containing .

**To prove**: .

**Proof**:

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

1 | Fact (1) | is -Sylow in | direct | ||

2 | Facts (1), (2) | is -Sylow in , | as given. Thus, by Fact (2), is a -Sylow subgroup of . Also, since , we have . Thus, applying fact (1) to the group , we get . | ||

3 | Fact (3) | Fact-direct | |||

4 | Steps (1), (2), (3) | Go mod in Step (3) and plug in Steps (1) and (2) into the right side to obtain the conclusion. |