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# Group theory question watch

1. If (G,*) is a group and e is the identity element in G, then I need to prove that e is in H (H is a finite subset of G) and (H,*) is also a group. Given that e is in G, how do I prove that it is also in H, and how do I prove that inverses are also in H?
2. What is H, a subset of G?
3. (Original post by kfkle)
What is H, a subset of G?
Yep, I should have mentioned that, I'll edit it now.
4. Are there any other restrictions for your subset? It does not necessarily follow that H contains the identity...
5. (Original post by kfkle)
Are there any other restrictions for your subset? It does not necessarily follow that H contains the identity...
Well on a summary sheet we've been given, it says,

"Let H be a subgroup of (G, *) and let e denote the identity element of G. Then e is in H and (H, *) is a group."

There are no other restrictions.
6. Well by definition H is a subgroup of G if H is a subset of G and H is a group with the restriction of the binary operation to the elements of H. So proving that H is a group is trivial: it's a group by definition. And proving that the identity of G is an element of H is pretty much immediate from this fact.
7. (Original post by nuodai)
Well by definition H is a subgroup of G if H is a subset of G and H is a group with the restriction of the binary operation to the elements of H. So proving that H is a group is trivial: it's a group by definition. And proving that the identity of G is an element of H is pretty much immediate from this fact.
Thanks for that, I temporarily got very confused about it but it's obvious now. Cheers!

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