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Groupprops β

Hall is transitive

Statement

Verbal statement

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

Facts used

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].