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I know that $a$ and $b$ are generators of a group $G$ and $a^{3^3}=b^9=1$.

  1. Are these informations sufficient to affirm that the group is a $3$-group?
  2. Adding the relation $b^{-1}ab=a^4$, can we state that it is a $3$-group?

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  1. No. We can just keep taking elements $ab$, $abab$, $ababab$, etc. and there is nothing in the relations to stop us. Without relating $a$ and $b$ somewhere, the group is infinite.
  2. Now, there is enough to tell that it is a $3$-group - in fact, it is the Sylow $3$-subgroup of the holomorph of $\mathbb{Z}_{27}$. This is a special case of the following characterization: whenever you have $\langle a,b | a^{p^3}=b^{p^2}=1, a^b=a^{p+1}\rangle$, this presentation describes the Sylow $p$-subgroup of the holomorph of $\mathbb{Z}_{p^3}$.
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    @Flast9 There's no absolute way which works for every presentation - it's like solving a puzzle. Sometimes group presentations can be very complicated. In general, I try to figure out the commutators between each generator, and see if you can deduce the order of every element.2012-10-29