In my question,

For group G which is the Semidirect product of $\mathbb{Z}_3$ and $\mathbb{Z}_7 \times \mathbb{Z}_7$, where $\mathbb{Z}_7 \times \mathbb{Z}_7$ is the normal subgroup, if z is a generator for the sylow 3-subgroup of G, then z induces an automorphism of $\mathbb{Z}_7 \times \mathbb{Z}_7$ via conjugation. That means, z induces an invertible linear transformation $T$ of $\mathbb{Z}_7 \times \mathbb{Z}_7$ such that $T^3=I$. Therefore minimal polynomial m(x) of $T$ divides $x^3-1 = (x-1)(x^2+x+1)$ and must be of degree atmost 2. So m(x)=$(x^2+x+1)$. Let {x,y} be a basis for $\mathbb{Z}_7 \times \mathbb{Z}_7$ and $x=T(y)=z^{−1}yz$. The matrix $\pmatrix{0 & -1 \\ 1 &-1}$ can be taken as one possibility for $T$. For the above matrix it can be observed directly that $T(ix+jy)=-jx+(i-j)y$.

I need to clarify a little bit more about how it can be observed that $T(ix+jy)=−jx+(i−j)y$. Is it by thinking $x=(1,0),y=(0,1)$ as standard basis and simplifying $T(ix+jy)$ ? Or is there another way? a more general way?

Thanks a lot in advance.

  • $\begingroup$ Yes, it is by identifying $x$ and $y$ with $(1,0)$ and $(0,1)$, and performing the matrix multiplication. $\endgroup$ – Nick Oct 10 '18 at 1:19
  • $\begingroup$ Thank you very much. $\endgroup$ – Buddhini Angelika Oct 10 '18 at 2:42

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