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Let $G$ and $H$ be two groups and let $f:G\longrightarrow H$ be an isomorphism.

Are the two following properties true?

1) If $g$ is a generator of $G$ then $f(g)$ is a generator of $H$ and more generally if $\langle g_1,\dotsc,g_n | R_G \rangle $ is a presentation of $G$, $R_G$ being a set of relations, then $\langle f(g_1),\dotsc,f(g_n) | f(R_G) \rangle $ is a presentation of $H$.

2) If $h$ is a generator of $H$, then $f^{-1}(h)$ is a generator of $G$.

  • 2
    Is this a homework question?2011-05-01
  • 0
    How are you defining a generator of a group? It is more usual to refer to a generating set. The second part of your first question is more in that vein. In any case the answer will be yes and yes! (as was said below).2011-05-01

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