1. ## Abstract Algebra 1

Let G be a group and let H and K be subgroups of G. Suppose there are
a, b exist in G such that Ha = Kb. Prove that H = K.
Hint: H and K are sets, so you must prove that H is a subset of  K and K is a subset of H.

Any help would be very helpful. I have tried countless ways to solve this but keep getting stuck! HELP PLEASE!

2. Originally Posted by nconti
Let G be a group and let H and K be subgroups of G. Suppose there are
a, b exist in G such that Ha = Kb. Prove that H = K.
Hint: H and K are sets, so you must prove that H is a subset of  K and K is a subset of H.

Any help would be very helpful. I have tried countless ways to solve this but keep getting stuck! HELP PLEASE!

Denote by 1 the group's unit, then with $\displaystyle h=1$ we get $\displaystyle 1\cdot a= kb$ , for some $\displaystyle k\in K\Longrightarrow ba^{-1}=k^{-1}\in K$ .

Now, let $\displaystyle h\in H$ be any element, then there exists $\displaystyle k'\in K\,\,\,s.t.\,\,\,ha=k'b\Longrightarrow h=k'(ba^{-1})=k'k^{-1}\in K\Longrightarrow h\in K\Longrightarrow H\subset K$ since

the last part was true for any element in H .

Now you prove the opposite direction in a similar way.

Tonio