Results 1 to 2 of 2

Math Help - Direct Product

  1. #1
    Junior Member
    Joined
    Aug 2007
    Posts
    32

    Direct Product

    Let G = (G, . , e), H = (H , * , E) be groups ... (e is the identity)
    the direct product is defined by:

    G x H = (GXH, o , (e,E)) where,
    (g1,h1) o (g2,h2) = (g1 . g2, h1*h2)

    Question: Show formally that G x H is a group.
    when it says, "show formally that G x H is a group".. does it mean show that G x H satisfies the 4 conditions for being a group.. i.e.

    associativity,
    closure,
    existance of identity element and
    existance of inverse element?

    any help is appreciated..
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by smoothman View Post
    Let G = (G, . , e), H = (H , * , E) be groups ... (e is the identity)
    the direct product is defined by:

    G x H = (GXH, o , (e,E)) where,
    (g1,h1) o (g2,h2) = (g1 . g2, h1*h2)

    Question: Show formally that G x H is a group.
    when it says, "show formally that G x H is a group".. does it mean show that G x H satisfies the 4 conditions for being a group.. i.e.

    associativity,
    closure,
    existance of identity element and
    existance of inverse element?
    First, closure is not a requirement. Because we define a binary operation * such that *:G\times G\mapsto G, so of course it is closed. However, a subgroup needs to satisfy the closure property.

    If you define G\times H)\times (G\times H)\mapsto G\times H" alt="*G\times H)\times (G\times H)\mapsto G\times H" /> as (g_1,h_1)*(g_2,h_2) = (g_1g_2,h_1,h_2) where g_1g_2 is the binary operation on G and h_1h_2 is the binary operation on H then * defines a binary operation.

    Proving the condition for G\times H to be a group are straightforward. For example, if you chose (e,e') where e is identity element in G and e' where e' is identity element in H. Then (g,h)*(e,e') = (ge,he')=(g,h). And similarly (e,e')*(g,h) = (g,h). Try to prove the other properties.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subgroups of the Direct Product
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 20th 2011, 11:33 PM
  2. Direct Product
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: January 16th 2011, 06:57 PM
  3. Direct Product
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 24th 2009, 08:10 PM
  4. Direct product
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: January 7th 2009, 08:41 PM
  5. External Direct Product
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 8th 2008, 04:29 PM

Search Tags


/mathhelpforum @mathhelpforum