Thread: wreath product of groups

Is $\displaystyle S_m \wr S_m \ldots \wr S_m$ (n times) $\displaystyle \cong S_{m^n}$ ? , where $\displaystyle \wr$ denotes the wreath product of groups.

plz explain.

Originally Posted by thippli

Is $\displaystyle

S_m \wr S_m \ldots \wr S_m$ (n times) $\displaystyle \cong S_{m^n}$ ? , where $\displaystyle \wr$ denotes the wreath product of groups.

plz explain.

no! compare the orders. for example for $\displaystyle m>1$ and $\displaystyle n=2$ we have $\displaystyle |S_m \wr S_m|=(m!)^{m+1} \neq (m^2)! =|S_{m^2}|.$

Thank you !