Math Help - wreath product of groups

1. wreath product of groups

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

plz explain.

2. Originally Posted by thippli
Is $

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

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

3. Thank you !