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In a Group S, ab!=ba, choose the possible relationship?

Posted: Sat Sep 22, 2012 4:35 pm
by 26186514
A. $$a^2=b^2$$
B.aba=1

Re: In a Group S, ab!=ba, choose the possible relationship?

Posted: Sat Sep 22, 2012 11:20 pm
by vonLipwig
To investigate whether the first one is possible, think of some small groups and try and find a and b with a^2 = b^2 and ab =/= ba.

For the second, manipulate the equation aba = 1 and see what happens.

Re: In a Group S, ab!=ba, choose the possible relationship?

Posted: Sat Sep 22, 2012 11:35 pm
by 26186514
Oh thanks! I see the second one now.
vonLipwig wrote:To investigate whether the first one is possible, think of some small groups and try and find a and b with a^2 = b^2 and ab =/= ba.

For the second, manipulate the equation aba = 1 and see what happens.

Re: In a Group S, ab!=ba, choose the possible relationship?

Posted: Sun Sep 23, 2012 3:26 am
by vonLipwig
Excellent. What did you get?

For the first one, try the symmetric group S_3.

Re: In a Group S, ab!=ba, choose the possible relationship?

Posted: Sun Sep 23, 2012 11:57 am
by 26186514
yes. Coz ab=ab(aba)=(aba)ba=ba if aba=1
and S3 is a brilliant idea! thanks!

Re: In a Group S, ab!=ba, choose the possible relationship?

Posted: Sun Sep 23, 2012 12:10 pm
by vonLipwig
Yep! Good work =)