Hall is transitive

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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., transitive subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about Hall subgroup |Get facts that use property satisfaction of Hall subgroup | Get facts that use property satisfaction of Hall subgroup|Get more facts about transitive subgroup property


Statement

Verbal statement

Any Hall subgroup of a Hall subgroup of a finite group, is a Hall subgroup in the whole group.

Facts used

  1. Index is multiplicative
  2. Lagrange's theorem

Proof

Given: A finite group G, subgroups H \le K \le G such that H is a Hall subgroup of K and K is a Hall subgroup of G.

To prove: H is a Hall subgroup of G.

Proof: By fact (1), we have:

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

Now, since H is Hall in K, the order of H is relatively prime to [K:H].

By fact (2), the order of H divides the order of K, and since K is a Hall subgroup of G ,the order of K is relatively prime to [G:K]. Thus, the order of H is relatively prime to [G:K].

Thus, the order of H is relatively prime to the product [G:K][K:H], which, by the above equation, equals [G:H].