Find the center of a specific group The group $G$ is generated by the two elements $\sigma$ and $\tau$, of order $5$ and $4$ respectively. We assume that $\tau\sigma\tau^{-1}=\sigma^2$.
I have shown the following:
* $\tau\sigma^k\tau^{-1}=\sigma^{2k}$ and $\tau^k\sigma\tau^{-k}=\sigma^{2^k}$.
* $\langle\sigma\rangle$ is a normal subgroup of $G$, and $\langle\sigma\rangle\cap\langle\tau\rangle=\{e\}$.
* $G/\langle\sigma\rangle=\langle\tau\langle\sigma\rangle\rangle$.
* $G$ is of order $20$ and every element $g$ in $G$ may be written uniquely in the form $g=\sigma^k\tau^m$, where $0\le k<5$ and $0\le m<4$.
* The commutator subgroup $[G:G]=\langle\sigma\rangle$.
What remains is to find the center $Z(G)$ of $G$. Any suggestions on how to proceed? Thank you.

I found $Z(G)=\{1\}$.
Here is my calculation.Let $K=<\sigma>$ and $H=<\tau>$
Since $H$ is cyclic, $H\leq N_G(H)$  and $N_G(H)\cap K=1$ otherwise, $K\leq N_G(H)\implies N_G(H)=G\implies $ $G$ is abelian contradiction.(if $H$ is normal,then $G\cong H\times K$)
Thus,we must have $N_G(H)=H\implies Z(G)\leq H$
Since $K$ is cylic $K\leq C_G(K)$ and since the other elements does not commute with $K\implies$ $ C_G(K)=K \implies Z(G)\leq K$ since $K\cap H=1$ hten we have $Z(G)=1$ .
