# Talk:Direct sum of groups

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Field:  Algebra

## Problems

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 the category of finite groups. In any case this article should redirect to Direct product of groups. Revolver 02:43, 21 July 2005 (UTC)

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 Gi = Z, the integers, for i = 1, 2, 3,... Then take the direct sum of the Gi 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 Gi, in the sense defined above, but it is not "isomorphic to a direct product of subgroups", since the direct product of the subgroups Gi 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 is 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)

My remark 02:43, 21 July 2005 (UTC) is correct. I mean, it should redirect to Free product of groups while this 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 the direct sum. Revolver 21:15, 31 August 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 other areas of math. And in any case the statement I mentioned above was still false, regardless. Revolver 05:01, 1 September 2005 (UTC)

## The lead is too long

Right now, the lead is much too long. Many details need to be moved from the lead into the body of the article. Bender2k14 (talk) 18:45, 1 May 2012 (UTC)

## An expert should have a look

I am not an expert in this field, but to me the article is confusing.

```* in other words, G is generated by the subgroups {Hi}.
```

Following the link one doesn't learn what it means that a group is "generated" by a set of groups. Probably what is meant is what is explained in the section "Generalization to sums over infinite sets" so that the definition there of the "external" direct sum should be taken as the actual definition of what the article is about.

```* Loosely speaking, a direct sum is isomorphic to a weak direct product of subgroups.
```

Qualifying the well understood concept of a direct product with the word "weak" does not bring any deeper insight to me. Nowhere else on Wikipedia one will find a hint as to how to understand "weak" in this context. Drop this sentence.

```* Thus, in a sense, the direct sum is an "internal" external direct sum.
```

In which sense? One can find on Wikipedia this definition of an "internal" direct sum, but it doesn't match the present use of the word.

Any expert, please have a look.

Kai Neergård (talk) 11:18, 5 March 2013 (UTC)

Added note: The reason why I want an expert to have a look is that I think the terminology on Wikipedia should be that of practitioners in the field. On the other hand, in my own field physics, in particular quantum mechanics, there is a long tradition of using the term "direct sum" just synonymously with "direct product" in relation to such objects as matrices and group representations. This seems to be also the viewpoint in the articles Direct sum and Matrix addition#Direct sum. Presently, some matematicians seem to want "direct sum" to be synonymous with the category-theoretical concept of a coproduct. Why actually, when this other term is there already? The aforementioned use of "direct sum" is much older than category theory.

Kai Neergård (talk) 00:43, 6 March 2013 (UTC)

## terminology

This article is sort of confused with three distinctions. One is just terminology. "Sum" seems more common for the abelian case versus "Product" for non-abelian. The second is internal versus external direct product/sum. The third is the infinite index set business, where the categorical sum (coproduct) and categorical product diverge.

I think these are all already covered more clearly on the direct product of groups page. I don't see what this article adds. I'd say just redirect it. --206.214.242.230 (talk) 16:17, 12 February 2015 (UTC) PS: The categorical coproduct for non-abelian groups is actually the free product.

New Author. I apologize if I should have started a new section. But IMO this article at the very least has functionality as a descriptor of how many authors use the term. IOW it may need more references as to where this may be true. — Preceding unsigned comment added by YouRang? (talkcontribs) 17:28, 3 July 2015 (UTC) E.g. if this is a correct description of what is meant on https://en.wikipedia.org/wiki/Finitely-generated_abelian_group by "direct sum of groups", then it is appropriate. If not, then perhaps this other page is the one that should be edited. — Preceding unsigned comment added by YouRang? (talkcontribs) 17:31, 3 July 2015 (UTC)