Hall satisfies transitivity

From Groupprops

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 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

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 $[G : H]$ .

Facts about Hall satisfies transitivityRDF feed

Fact about	Hall subgroup +, and Transitive subgroup property +
Uses	Index is multiplicative +, and Lagrange's theorem +

Hall satisfies transitivity

From Groupprops

Contents

Statement

Verbal statement

Facts used

Proof

Views

Personal tools

Search

Navigation

lookup

Credits

request/feedback

Toolbox