# Request for clarification about a computation in semidirect products

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?

• Yes, it is by identifying $x$ and $y$ with $(1,0)$ and $(0,1)$, and performing the matrix multiplication. – Nick Oct 10 '18 at 1:19