General formula for exponent of a semidirect product

Question

This page states that

Exponent of semidirect product may be strictly greater than lcm of exponents

But it doesn't give any proof for that.

Could anyone provide a general formula for the exponent of a semidirect product?

Answer 1

The simplest example is probably the group $(\mathbf C_2 \times\mathbf C_2)\rtimes\mathbf C_2\cong\mathbf D_8$. The action is given by swapping the components. You can see that it must be $\mathbf D_8$ since, for instance, it is a non-abelian group of order $8$, which leaves only $\mathbf D_8$ and $\mathbf Q_8$, with a subgroup isomorphic to $\mathbf C_2\times\mathbf C_2$, leaving only $\mathbf D_8$.

Then $\mathrm{exp}(\mathbf C_2\times\mathbf C_2)=2$, $\mathrm{exp}(\mathbf C_2)=2$ and their lcm is $2$, while $\mathrm{exp}(\mathbf D_8)=4$.

This example may be generalized to $(\mathbf C_p)^p \rtimes \mathbf C_p$ (also known as the wreath product $\mathbf C_p\wr\mathbf C_p$). This group $\mathbf C_p\wr\mathbf C_p$ is also notorious for being the Sylow $p$-subgroup of $\mathbf S_{p^2}$. So certainly there will be an element of order $p^2$ in it!