Thread: 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!

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 we get , for some .

Now, let be any element, then there exists since

the last part was true for any element in H .

Now you prove the opposite direction in a similar way.

Tonio