On wreath product of finite $p$-groups

Question

Actually, I was studying about the group $ C_p \wr C_p$, where $C_p$ is cyclic group of order $p$ and $\wr$ denotes the wreath product. I understood that it is isomorphic to the Sylow $p$-Subgroup of $S_{p^2}$. I was wondering if I can get a subgroup inside $GL(n,p^k)$, for some $k\in \mathbb{N}$ which is isomorphic to $C_p \wr C_p$?

Answer 1

One may find $P\cong C_p\wr C_p$ in the group $\mathrm{GL}_{2p}(p)$. The construction is fairly reasonable.

First, the Sylow $p$-subgroup of $\mathrm{SL}_2(p^p)$ is elementary abelian of order $p^p$. The field automorphism of order $p$ on $\mathrm{SL}_2(p^p)$ acts on this Sylow $p$-subgroup, and constructs exactly your group, $C_p\wr C_p$. So now we want a faithful representation of $G=\mathrm{SL}_2(p^p)\rtimes C_p$. I claim there is a $2p$-dimensional representation of this group over $\mathbb{F}_p$.

To see this, notice that $\mathrm{SL}_2(p^p)$ has $p$ different (faithful) representations of dimension $2$. Its natural representation is one, and then its twist under the Frobenius map, obtained by mapping $x\in \mathbb F_{p^p}$ to $x^p$. This is a field automorphism, and gives us $p$ different representations. The sum of all of these is of dimension $2p$, and is stable under the Frobenius map, by construction. This means that it is conjugate in $\mathrm{GL}_{2p}(p^p)$ to a subgroup of $\mathrm{GL}_{2p}(p)$.

The final bit is to note that the field automorphism of $\mathrm{SL}_2(p^p)$ still normalizes this subgroup of $\mathrm{GL}_{2p}(p)$, and so we find all of $C_p\wr C_p$ as a subgroup of this group.

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If one prefers cross-characteristic (i.e., over fields of characteristic different from $p$) representations, then one may do better than $\mathrm{GL}_{p^2}(\mathbb{C})$, and it lies in $\mathrm{GL}_p(\mathbb{C})$. One sees this by simply inducing a linear character from the subgroup of index $p$.