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  
Senior Member Joined: Aug 2012 Posts: 2,306 Thanks: 706  Re: direct product Quote:
(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: where each of these is the identity element in their respective groups. 

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