# Group theory

Please help with the following problems involving group theory proofs. Provide step by step calculations along with explanations.

a) Prove that if G is a finite group and a is an element of G then for some positive m , a^m is equal to the identity of G. (Use the Pigeon hole principle)

b) Prove that if G is a finite group, H subset of G that is closed with respect to the operation of G, Then every element of H has its inverse in H.

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Please see below for your solutions.

Proof:

(a) Prove that if G is a finite group and a is an element of G then for some positive m , a^m is equal to the identity of G. (Use the Pigeon hole principle)

For any element a in G, since G is finite, we can assume |G| = n, ...

#### Solution Summary

The following posting helps with a problem regarding group theory, and involves providing proofs. This provides examples of group theory proofs using pigeonhole principle and inverse elements. It helps prove that a group is a finite group based on the two types of proofing principles.