need more help ...

**deniselim17** · November 1st 2008, 02:58 AM

$G_{p}$ 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.

**deniselim17** · November 1st 2008, 05:27 PM

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 ?

**ThePerfectHacker** · November 1st 2008, 07:17 PM

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 $G_p\subseteq H_p$ .

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