Representatives of conjugacy classes of the wreath product $\mathbb{Z}_p \wr \mathbb{Z}_p$

Question

Let $p$ be a prime number and $G$ be the wreath product of the cyclic group with itself $\mathbb{Z}_p \wr \mathbb{Z}_p\cong (\mathbb{Z}_p)^p \rtimes \mathbb{Z}_p$. What I want to know is if there is an explicit set of representatives for the conjugacy classes of this group.

I read this question Determining the conjugacy classes of a wreath product $G \wr S_n$ but the answer fails to answer my questions since it does not express clearly which set of representatives I can take.

I am mostly interested in a set of representatives of the subgroup $(\mathbb{Z}_p)^p$ (it is normal in $G$), since I have already a set of representatives for elements outside of this subgroup.

Answer 1

Identify the elements of $(\mathbb Z _p) ^p$ with the $p$-dimensional vector space on $\mathbb Z_p$, such that the elements of $(\mathbb Z _p) ^p$ are represented as $(x_1,x_2,...,x_p)$ where $x_k \in \mathbb Z_p$ for $k \in [1..p]$.

Claim. $x,y$ in $(\mathbb Z _p) ^p$ are conjugate in $G$ ⇔ $y$ is a cyclic shift of $x$.

Proof. $(\Leftarrow)$ obvious.

$(\Rightarrow)$ It follows that $y=axa^{-1}$ where $a\in G$. $a$ can be decomposed into $qr$, where $q$ is a cyclic shift and $r$ is in $(\mathbb Z _p) ^p$. Thus

$y=qrxr^{-1}q^{-1}=qxq^{-1}$ , so y is a cyclic shift of $x$.

So an explict set of representatives of conjugacy classes of $(\mathbb Z _p) ^p$ in $G$ can be generated by the following algorithm:

C={}
for x in (ℤₚ)ᵖ:
    if x is lexicographically the first among the cyclic shifts of x:
        C=C∪{x}
    end
end

More sophisticated algorithms such as canonization may be used for faster enumeration.