Results 1 to 3 of 3

Math Help - need more help ...

  1. #1
    Member
    Joined
    Mar 2008
    Posts
    85

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Mar 2008
    Posts
    85
    Quote Originally Posted by deniselim17 View Post
    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 ?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    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 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.
    Follow Math Help Forum on Facebook and Google+


/mathhelpforum @mathhelpforum