Hall satisfies intermediate subgroup condition

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

Any Hall subgroup of a finite group is also a Hall subgroup in any intermediate subgroup.

Related facts

Facts used

  1. Index is multiplicative

Proof

Given: A finite group G, a Hall subgroup H, and a subgroup K of G containing H.

To prove: H is a Hall subgroup of K.

Proof: Note that by the multiplicativity of index:

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

Thus, the index [K:H] divides the index [G:H]. In particular, if |H| and [G:H] are relatively prime, so are |H| and [K:H].