in an exercise , i'm given a definition of restricted direct product , but it doesn't make sense for me !
they say , the restricted direct product of groups $G_i$ is the set of elements of direct product which are the identity in all but finitely many compontents
i really don't understand what it means to be all but finitely !
i googled and found the same defintion with No clearification !
so , any help by example or explanation ?
Best Answer
Note first that when we are taking the product of a finite number of groups, then the restricted and normal direct product are the same.
For convenience, let us assume that we have a family of groups indexed by the natural numbers, so we have a $G_i$ for each $i\in\mathbb{N}$.
The usual direct product of the groups $G_i$ can be seen as the set of sequences such that the $i$'th entry of the sequence is an element of $G_i$ (we can also write these as infinite tuples $(g_1,g_2,\dots)$ with each $g_i\in G_i$).
The restricted direct product is then the subgroup of the above direct product where we only take those elements such that for some $N$, the $i$'th entry is $e_i$ whenever $i\geq N$ (I use $e_i$ to denote the identity element of $G_i$). This means that we can see the restricted direct product as the set of finite tuples $(g_1,g_2,\dots,g_n)$ (but where $n$ can be as large as we want) and where we multiply such tuples by first extending the shortest of them by adding $e_i$'s to the end, such that they get the same length, and then multiply them as we usually would do with tuples (and we identify two such tuples if the only difference between them is how many $e_i$'s they end in).