Proving Subgroup Equality in Group Theory

[ranoo]

Homework Statement

let H a subgroup of G. Prove that:
1) Ha=H iff a belong to H
2) H²=H

Homework Equations

The Attempt at a Solution

 

[Mark44]

Homework Statement

let H a subgroup of G. Prove that:
1) Ha=H iff a belong to H
2) H²=H

What does Ha mean? What does it mean to multiply a subgroup by itself?

 

[ranoo]

I don't know, like this we took in the homework.

 

[quenderin]

Ha is the set containing all elements of the form h*a, where h is in H.

What is the definition of a subgroup? It must non-empty, closed under multiplication and closed under inversion. So if Ha = H, then h_{1}a = h_{2}. What does this say about a?

If a is in H on the other hand, and H is closed under multiplication, what does this say about Ha?

Finally, to prove that H^2=H, we need to show that H^{2} \subseteq H and H \subseteq H^{2}. First, naively, which one is definitely contained in which? Next, if H is closed under multiplication, what does this say about the relation between H and H^2?

Good luck!

 

[ranoo]

thank you very much