I came across the following question.
Consider the semidirect product $G=(\mathbb{Z}_p \times \mathbb{Z}_p)\rtimes \mathbb{Z}_3$, where $p(>3)$ is a prime. Also it is assumed that the commutator subgroup, $[G,G]=\mathbb{Z}_p \times \mathbb{Z}_p$. Let $G=<s,t>$, where $|s|=3,\, |t|=p$ and an action of $s$ on $\mathbb{Z}_p \times \mathbb{Z}_p$ is defined as follows.
-Define a linear transformation $T$ on $\mathbb{Z}_p \times \mathbb{Z}_p$ by $T(v)=s^{-1}vs$,
-let $m(x)$ be the minimal polynomial of $T$ and
-Let $u=T(t)=s^{-1}ts$
Note that:
- because ${s}=3$, we know that $T^3=I$, so $m(x)$ divides $x^3-1=(x-1)(x^2+x+1)$
- since $|[G,G]|=p^2$, we know that 1 is not an eigen value of $T$
- because $<s,t>=G$, we know $u=T(t) \notin <t>$, so the minimal polynomial of $T$ ha degree 2 (and $\{t,u\}$ is a basis of $\mathbb{Z}_p \times \mathbb{Z}_p$).
With respect to the point 1. I thought if $\phi : \mathbb{Z}_q \rightarrow Aut(\mathbb{Z}_p \times \mathbb{Z}_p)$ (the homomorphism corresponding to the semidirect product), then $T$ is like an element of the $Aut(\mathbb{Z}_p \times \mathbb{Z}_p)$. Then it is like,
$\phi(s) = T$, where $T(v)=s^{-1}vs$,
$\phi(s^2) = \phi(s*s) = \phi(s) \phi(s) =T^2$,
$\phi(s^3)=\phi(e)=I$ and also $\phi(s^3)=\phi(s*s*s)=\phi(s) \phi(s) \phi(s)=T^3$ and so we can say $T^3=I$. And also I have read in some text that "Let $A$ be a linear transformation of finite order on a complex vector space: $A^m=I$, for some positive integer $m$. Since $A^m=I$, $A$ is killed by $\lambda^m-1$. Therefore, the minimal polynomial is a factor of $\lambda^m-1$". So I determined that saying $m(x)$ divides $x^3-1$ comes from this argument (but I'm bit unclear since it says complex vector space for $A$ anyway) Am I right?
But the 2. and 3. points are not very clear to me. How can we determine that 1 can't be an eigen value based on the order of commutator subgroup? What is the importance of commutator subgroup for the linear transformation, eigen values and eigen vectors?
And in point 3, how did they determine that minimal polynomial should have degree 2 and $\{t,u\}$ must be a basis.
Can someone please help me to understand these arguments.
Thanks a lot in advance.