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Let $G=\langle a,b|a^2=b^3=1,(ab)^n=(ab^{-1}ab)^k \rangle$. Prove that $G$ can be generated with $ab$ and $ab^{-1}ab$. And from there, $\langle(ab)^n\rangle\subset Z(G)$.

Problem wants $H=\langle ab,ab^{-1}ab \rangle$ to be $G$. Clearly, $H\leqslant G$ and after doing some handy calculation which takes time I've got:

  1. $ab^{-1}=(ab^{-1}ab)(ab)^{-1}\in H$

  2. $b=b^{-2}=(ab)^{-1}ab^{-1}\in H$

  3. $a=(ab)b^{-1}\in H$

So $G\leqslant H$ and therefore $G=H=\langle ab,ab^{-1}ab\rangle$.

For the second part, I should prove that $N=\langle(ab)^n\rangle\leqslant Z(G)$.

Please help me.

Thanks.

  • 0
    @BabakSorouh, Thanks. "Presentation of groups". Finding Forms of Algebraic objects is always amusing and nice.2013-01-09

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