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 January 21st, 2014, 09:14 AM #1 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 direct product How can I prove that direct product of two groups is group? Or it did not?
 January 21st, 2014, 11:52 AM #2 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: direct product Depending on how you define the binary operator.
 January 22nd, 2014, 10:20 AM #3 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 Re: direct product I need to determine a direct prodcut binary operation that operate and if is possible direct product that doesn't operate or vice versa. Tell me - I don't know how to do it. May one look on it and tell me how to deal with it. Thanks, for the responders.
 January 22nd, 2014, 10:47 AM #4 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: direct product If you are given two groups with binary operations (G, g) and (H, h) use the binary operator over G x H as (g, h). You have a group then.
 January 22nd, 2014, 12:05 PM #5 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 Re: direct product I don't understand which opeartion you decide to use? What is the operation you choose? I think you choose operation of Cartesain product? But what is the result? (I think it's a point). Please someone expand and explain the operation that operate here. Thanks and Bless you.
January 22nd, 2014, 05:30 PM   #6
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Re: direct product

Quote:
 Originally Posted by shaharhada I don't understand which opeartion you decide to use? What is the operation you choose? I think you choose operation of Cartesain product? But what is the result? (I think it's a point). Please someone expand and explain the operation that operate here. Thanks and Bless you.
Each of G and H are groups. So you combine pairs (g, h) and (g', h1') using the respective group operations on G and H. In other words

(g, h) * (g', h1') = (gg', hh')

where gg' is defined by the group operation on G; and hh' is defined by the group operation on H.

 January 23rd, 2014, 10:28 AM #7 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 More question: I have a more question: What is the identity element of the operating group result? Thanks for the guys who help me. How I find the identity element?
 January 23rd, 2014, 06:34 PM #8 Senior Member   Joined: Mar 2012 Posts: 294 Thanks: 88 Re: direct product The identity of GxH is the pair: $(e_G,e_H)$ where each of these is the identity element in their respective groups.

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