# Set Containment Problem

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• Apr 16th 2008, 05:30 AM
GoldendoodleMom
Abstract Algebra Problem
I see there is an old string addressing this question, but I need to prove it using set containment both directions. Not sure where to start.

a, b are in G, a group. H is a subgroup. If aH = bH then Ha^(-1) = Hb^(-1).
• Apr 16th 2008, 06:51 AM
Isomorphism
Quote:

Originally Posted by GoldendoodleMom
I see there is an old string addressing this question, but I need to prove it using set containment both directions. Not sure where to start.

a, b are in G, a group. H is a subgroup. If aH = bH then Ha^(-1) = Hb^(-1).

You can do this easily if you write the conditions clearly and think reverse.

To prove: $\displaystyle Ha^{-1} \subset Hb^{-1}$
That is to prove: $\displaystyle \forall h \in H, ha^{-1}b \in H$
But that is the same as proving $\displaystyle a^{-1}b \in H$
Now since, $\displaystyle aH = bH \Rightarrow a^{-1}b \in H$, we are done.
The other way round is exactly similar except for different variable name.