Results 1 to 3 of 3

Thread: need more help ...

  1. #1
    Member
    Joined
    Mar 2008
    Posts
    96

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Mar 2008
    Posts
    96
    Quote Originally Posted by deniselim17 View Post
    $\displaystyle G_{p}$ 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 ?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by deniselim17 View Post
    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 $\displaystyle G_p\subseteq H_p$.
    Follow Math Help Forum on Facebook and Google+


/mathhelpforum @mathhelpforum