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February 3rd, 2011, 12:34 PM   #1
 
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Show that (ab)^n=(a^n)(b^n) for an Abelian group

I'm having problem with this proof: Let a and be be elements of an Abelian group and let ne be any integer. Show that (ab)^n=(a^n)(b^n). Is this true for non Ablelain groups. I am two weeks in to my course in Abstract Algebra and I have an exam need week. Really nned help because I am gettine myself so confused!
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February 3rd, 2011, 01:22 PM   #2
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Re: Show that (ab)^n=(a^n)(b^n) for an Abelian group

Well, what is (ab)^n ??
It's just (ab)(ab)(ab)(ab).........(ab)
n times.
But since you're talking abelian, you can move the elements around, so you can write
(aaaaaaaaaaaaaaaa.........a)(bbbbbbbbbbbbbbbbbb... .....b)
Where each of a and b is written n times, but that's just
(a^n)(b^n)

This is not true in non-abelian groups.
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February 3rd, 2011, 02:53 PM   #3
 
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Re: Show that (ab)^n=(a^n)(b^n) for an Abelian group

Thats bascially what i had only i thought it was too simple!!
Thanksvery much!
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February 3rd, 2011, 02:55 PM   #4
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Re: Show that (ab)^n=(a^n)(b^n) for an Abelian group

You can also (repeatedly) multiply by a^-1 on the left and b^-1 on the right....
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February 3rd, 2011, 03:23 PM   #5
 
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Re: Show that (ab)^n=(a^n)(b^n) for an Abelian group

But the first option is still ok for a proof right??
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February 3rd, 2011, 03:40 PM   #6
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Re: Show that (ab)^n=(a^n)(b^n) for an Abelian group

Quote:
Originally Posted by Elladeas
But the first option is still ok for a proof right??
It's still a proof. If you're expected to be more formal than that, use induction to show it works.
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February 5th, 2011, 04:23 AM   #7
 
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Re: Show that (ab)^n=(a^n)(b^n) for an Abelian group

Ok thanks Ill try that
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