Originally Posted by

**deniselim17** $\displaystyle G_{p}$ is the set of all elements in $\displaystyle G$ whose order is a power of $\displaystyle p$. The subgroups $\displaystyle G_{p}$ of $\displaystyle G$ are called the *primary components* of $\displaystyle G$.

1. Let $\displaystyle G$ and $\displaystyle H$ be finite abelian groups, and let $\displaystyle f: G \longrightarrow H$ be a homomorphism. Show that $\displaystyle f(G_{p}) \subset H_{p}$, for each $\displaystyle p$.

2. Let $\displaystyle G$ and $\displaystyle H$ be finite abelian groups; show that $\displaystyle G \cong H$ if and only if $\displaystyle G_{p} \cong H_{p}$ for all primes $\displaystyle p$.

3. Let $\displaystyle G$ be a finite abelian group. Show that any two decompositions of G into direct sums of primary cyclic groups have the same number of summands of each order.