fundamental theorem of abelian groups ? Herstein I was reading Fundamental theorem of abelian groups in Herstein Second Edition
(pg 110), it was mentioned $b_2^{p^{n_2}} \in A_1$ and is the first power of $b_2$ to fall in $A_1.$ I was not able follow this, could someone suggest why it happens so ?
3rd paragraph, 7th line

The element $b_2$ is such that its image $\overline{b_2}=b_2+A_1$ in $G/A_1$ has maximal order $p^{n_2}$ i.e. none of the elements in $G/A_1$ has order more than $p^{n_2}$. So $(\overline{b_2})^{p^{n_2}}=A_1$ (identity element of $G/A_1$) $\Rightarrow b_2^{p^{n_2}}+A_1= A_1\Rightarrow b_2^{p^{n_2}}\in A_1$. Also, for any number $k$ with $k<p^{n_2}$, $b_2^k\notin A_1$ because otherwise $(\overline{b_2})^k=b_2^k+A_1=A_1$ which contradicts that $\overline{b_2}$ was an element of order $p^{n_2}$ in $G/A_1$.
