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 May 7th, 2011, 05:18 AM #1 Newbie   Joined: May 2011 Posts: 1 Thanks: 0 external direct product How do you prove that the external direct product of any finite number of groups is a group?
 May 7th, 2011, 08:43 AM #2 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 Re: external direct product Lila, 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 $G\=\ G_1\times G_2\times\cdots\times G_n,$ and $^{g,h,k \in G}$ are written as \begin{align*} g\ &=\ \left(g_1,g_2,\dots,g_n\right),\\ h\ &=\ \left(h_1,h_2,\dots,h_n\right),\\ k\ &=\ \left(k_1,k_2,\dots,k_n\right), \end{align*} so that the operation is defined by $gh\=\ \left(g_1h_1,g_2h_2,\dots,g_nh_n\right),$ then you need to show that $1)\quad (gh)k\ =\ g(hk),\\ 2)\quad \exists e \in G \text{ s.t. } eg\ =\ ge\ =\ g, \text{ existence of identity---the choice for it is the natural one}\\ 3)\quad \forall g,\ \exists g^{-1} \text{ s.t. } gg^{-1}\ =\ g^{-1}g\ =\ e.$ -Ormkärr-

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# how to show that the external direct product of groups is itself a group

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