Semidirect Product Definition of Addition

Question

Hello Everyone, I'm having a hard time trying to define the addition on this semidirect product, any suggestion would be appreciated.

Thanks.

Answer 1

We have: $(\mathbb{Z}_5 \times \mathbb{Z}_5)\rtimes_{\phi}\mathbb{Z}_3$ and $\phi:\mathbb{Z}_3\longrightarrow{\rm GL}_2(5).$ Now Define: $A =\left(\begin{array}{rr}0&1\\-1&-1\end{array}\right)$, since $A \in {\rm GL}_2(5)$ has order 3, then we can let: $\phi(1) = A$. Now, to calculate $x = (1,0,1)$, notice that:$$(h_1,k_1)(h_2,k_2) = (h_1\phi(k_1)(h_2),k_1k_2)),$$

So, if $h_1 =(1,0)$ and $k_1 = 1$, then $x =(h_1,k_1)$ and we compute: $$2x=(h_1,k_1) +(h_1,k_1) = ((1,0)+ A\left(\begin{array}{rr}1\\0\end{array}\right),2),$$ Therefore:$$2x=((1,-1),2).$$ Similarly we can do the same with $3x = 2x+x.$