I'm trying to understand a proof to the Cauchy's Theorem mentioned above, and this particular proof I'm reading, from Herstein, claims at some point that given $G$ a finite abelian group and $N$ a subgroup of $G$, if:
- $p \nmid o(N)$, where $p$ is a prime
- $(Nb)^{o(N)} = N$, where $b$ is an element of $G$ and $o(N)$ denotes the order of $N$.
- $(Nb)^p$ = N
Then $Nb = N$
If that really is the case, could you explain to me why, please?
Thanks in advance!