Sylow satisfies intermediate subgroup condition

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This article gives the statement, and possibly proof, of a subgroup property (i.e., Sylow subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
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Statement

A Sylow subgroup of a finite group is also a Sylow subgroup in any intermediate subgroup.

Definitions used

Sylow subgroup

Further information: Sylow subgroup

A subgroup P of a group G is a Sylow subgroup if P is a group of prime power order and the order of P is relatively prime to the index of P.

P is termed a p-Sylow subgroup if its order is a power of the prime p and its index is not a multiple of p.

Related facts

Facts used

  1. Index is multiplicative

Proof

Given: A finite group G, a p-Sylow subgroup H of G, a subgroup K of G containing H.

To prove: H is a Sylow subgroup (in fact, a p-Sylow subgroup) inside K.

Proof: Clearly, H continues to be a p-subgroup inside K, because it is a p-group. Thus, it suffices to show that the index [K:H] is relatively prime to p. For this, observe that by fact (1):

[G:H] = [G:K][K:H]

Thus, [K:H] is a divisor of [G:H]. Since [G:H] is relatively prime to p, so is [K:H].