(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let G=<a, b|a^2=b^2=e, aba=bab>. SHow G is isomorphic to S3.

2. Relevant equations

3. The attempt at a solution[/

since a^2=b^2=e, then |a|=1 or 2 and |b|= 1 or 2. But since aba=bab, the orders of a and b both have to be 2 because if either had order 1, we would get that a=a^2 or that b=b^2. Since aba= bab, by left hand multiplication and right hand multiplication by ab and (ab)^-1 we get that (ab)(aba)(ab)^-1 = abab(ab)^-1= ab. So then ab and aba are conjugates, from this we can partion G by <aba>. So G= a<aba> union b<aba> union e<aba>. Thus G has at most 6 elements (since |aba|=2). SO |G|<= 6. From this, since |S|>=|G| we can show that S3 satisfies the defining relations of G.

Is my thinking on this right?

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# S3 question

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