# Thread: need more help ...

1. ## need more help ...

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

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

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

3. Let $G$ 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.

2. Originally Posted by deniselim17
$G_{p}$ is the set of all elements in $G$ whose order is a power of $p$. The subgroups $G_{p}$ of $G$ are called the primary components of $G$.

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

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

3. Let $G$ 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.
I know how to show no. 2, but no. 1 and no. 3 I don't have any idea how to show them.
Can anyone give me some hint in showing no. 1 and no. 3 ?

3. Originally Posted by deniselim17
I know how to show no. 2, but no. 1 and no. 3 I don't have any idea how to show them.
Can anyone give me some hint in showing no. 1 and no. 3 ?
How did you do #2 converse? Did you use the fact that if $G$ is finite abelian then $G \simeq C_1\times ... \times C_r$ where $C_i$ are cyclic groups of prime-power order. For #1 prove that if $f:G\to H$ is a group homomorphism then $|f(a)|$ divides $|a|$. It should follow then that if $f:G\to H$ is a group homomorphism then $G_p\subseteq H_p$.