Group theory problem

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Kurisu

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Nov 13, 2020
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Hey everyone! I'm struggling a bit with this group theory problem on my applied modern algebra homework. Anyone have any thoughts?

Let H be a subgroup of G and suppose that Ha = bH for a, b ∈ G. Show that aH = Hb.

I've been working on this for hours with almost no progress made. I feel like Lagrange's theorem is supposed to help me but I'm not sure how to apply it...
 
Can you please state LaGrange's theorem?

Can you post the work you are done so we know how you want to proceed. We then can offer some hints.

What element do you know is in Ha? Does that element have to be in bH? What does that tell you?
 
Jomo said:
Can you please state LaGrange's theorem?

Can you post the work you are done so we know how you want to proceed. We then can offer some hints.

What element do you know is in Ha? Does that element have to be in bH? What does that tell you?
Click to expand...
In Ha, I suppose element a must exist right?
 
Yes, since e is in H (it is a subgroup) we have a=ea is in Ha. Similarly b is in bH. Since Ha = bH we have both a and b in Ha and bH.

Now multiply Ha = bH on the right side by a-1 and on the left side by b-1. What does that tell you and how can you get aH = Hb from there?
 
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