# Thread: need more help ...

1. ## need more help ...

$\displaystyle G_{p}$ 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.

2. 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.
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 $\displaystyle G$ is finite abelian then $\displaystyle G \simeq C_1\times ... \times C_r$ where $\displaystyle C_i$ are cyclic groups of prime-power order. For #1 prove that if $\displaystyle f:G\to H$ is a group homomorphism then $\displaystyle |f(a)|$ divides $\displaystyle |a|$. It should follow then that if $\displaystyle f:G\to H$ is a group homomorphism then $\displaystyle G_p\subseteq H_p$.