Congruence condition on index of subgroup containing Sylow-normalizer
- 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
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Given: A finite group , a prime , a -Sylow subgroup , a subgroup of containing .
To prove: .
|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 .|
|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.|