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May 7th, 2011, 05:18 AM   #1
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external direct product

How do you prove that the external direct product of any finite number of groups is a group?
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May 7th, 2011, 08:43 AM   #2
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Re: external direct product


The first thing is to define what the operation on the product is going to be. Of course, it is what you would expect---component-wise addition (or multiplication, depending on what notation you are using). The second is to show that this operation makes the set (product) closed under the operation, etc., i.e. it satisfies all the group axioms. To that end, if

and are written as

so that the operation is defined by

then you need to show that

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