Talk:Direct sum of groups

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Mathematics rating:	Start Class	Mid Priority	Field: Algebra
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Needs references. Geometry guy 15:40, 10 June 2007 (UTC)

I believe what is described here is more accurately called the internal direct product of groups. The direct sum of groups occurs when you take the direct product and restrict so that all but finitely many coordinates are zero (identity). I personally dislike this terminology, because it implies that the direct sum construction is a coproduct in the category of groups, this is not even true for category of finite groups. In any case, this article should redirect to direct product of groups Revolver 02:43, 21 July 2005 (UTC)

Correct, I mean it should redirect to free product of groups while this the content should redirect to internal direct product of groups. Revolver 18:35, 30 August 2005 (UTC)

Another correction, I believe this is the internal weak direct product. Which is not the same as direct sum. Revolver 21:15, 31 August 2005 (UTC)

[edit] problems

Excerpt:

In group theory, a group G is called the direct sum of a set of subgroups {Hi} if

   * each Hi is a normal subgroup of G
   * each distinct pair of subgroups has trivial intersection, and
   * G = <{Hi}>; in other words, G is generated by the subgroups {Hi}.

If G is the direct sum of subgroups H and K, then we write G = H + K; if G is the direct sum of a set of subgroups {Hi}, we often write G = ∑Hi. Loosely speaking, a direct sum is isomorphic to a direct product of subgroups.

This is incorrect as stated. Take the case G_i = Z, the integers, for i = 1, 2, 3,... Then take the direct sum of the G_i as abelian groups, meaning each element has all but finitely many zero coordinates. Then this group, call it G, is indeed the internal direct sum of the G_i, in the sense defined above, but it is not "isomorphic to a direct product of subgroups", since the direct product of the subgroups G_i consists of all elements of the cartesian product, not just those with finitely many non-zero coordinates. This direct product is not isomorphic to G.

The article seems to be attempting to discuss internal direct sums and internal direct products, but it confuses the two. This problem is compounded by the fact that the assumption is constantly made that there are a finite number of groups. Although the direct sum and direct product are the same as objects here, they are not the same spiritually: the direct product of a finite number of abelian groups is the direct product group together with the projection maps, while the direct sum of a finite number of abelian groups is the direct product (= sum) group together with the injection maps. For general groups, it's different still: the direct product of groups is still the direct product with projections, but the "direct sum" is now the free product. All these things are confused together in the article. Revolver 02:59, 21 July 2005 (UTC)

I've changed my view somewhat. I'm willing to live with the current name, but I still think a strong warning is needed that this is not the usual "direct sum" idea as it usually is meant in most every other area of math. And in any case, the statement I mentioned above was still false, regardless. Revolver 05:01, 1 September 2005 (UTC)