Talk:Subquotient

Missing definition

The article talks about everything, except that what makes a group the subquotient of another. Probably it is some "part-of" relation, like the subgroup, but there is no way to know, exactly wtf. 188.195.226.36 (talk) 04:47, 10 May 2023 (UTC)

I read in the article:

"a subquotient is a quotient object of a subobject"

with links to the articles "quotient object" and "subobject".

If this is not sufficient for you, pls add for your convenience. –Nomen4Omen (talk) 19:19, 11 May 2023 (UTC)

Both links refer to the same article, and the definition itself says, it is not enough here. First it states, "In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category.", then "An appropriate categorical definition of "subobject" may vary with context, depending on the goal. One common definition is as follows.". Thus, the meaning is only that the "happy family" groups are not subgroups of the monster group, but they relate to it similarly.

That is clearly not enough. 188.195.226.36 (talk) 08:55, 3 January 2024 (UTC)

Sorry, I can't share your problems with this definition. As worked out in the article, it is even possible to prove for groups the transitivity of the subquotient relation. And as I already said on my previous response to your inquiry, so far nobody tries to hinder any attempt on your side to say more.

As far as I understand WP, the way of solving problems with an article is to read literature pertaining to the context and then possibly making improvements to the article.

On the other hand, you have extremely well understood that the "happy family" groups are not subgroups of the monster group, but they are subquotients of it. Nomen4Omen (talk) 10:14, 3 January 2024 (UTC)

@188.195.226.36:Could you pls have a look on my new example in the article. It is the wrong way around, namely its quotient is a subobject. But I am sure you will find a quotient group of a subgroup which is not a subgroup. Nomen4Omen (talk) 15:33, 3 January 2024 (UTC)

Too generally stated result

@Nomen4Omen: You recently added the unqualified claim that taking subquotients be an order relation to the article. I think that this is quite incorrect; the purposed proof definitely is. For this proof to work, you should make a similar restriction to finite objects as you did here, on dewp.

Actually, not even this seems to be quite without problems:

1. It essentially assumes that all considered categories are concrete, doesn't it? If not the objects in a category are sets and its morphisms certain functions between the sets, then I do not even understand what the cardinality of an object should be.
2. There is also the tacit understanding that you work with isomorphism classes rather than with the original objects; c. f. your "Bemerkung" wisely added here, but later removed.

If you want to discuss this, we can do it here, or at die Diskussionsseite des Artikels de:Subquotient, whichever you prefer.

Thus, indeed, the isomorphism classes of the full subcategory of finite groups of the category indeed form a partially ordered set; and this is a rather important situation, well worth to include. However, you should rework the claim so that it does not appear to hold in greater generality than it does. (I even doubt that the transitivity holds in a category without limits.) I believe you are the most suitable person to do this. Regards, JoergenB (talk) 15:36, 18 June 2024 (UTC)

@JoergenB: For me the biggest problem is the missing reference of literature – for both, the statement "order relation" and the "proof of transitivity for groups". It would be very helpful if you would find and add some – and maybe even upgrade the proof so that it works also for modules. Regards, Nomen4Omen (talk) 08:01, 21 June 2024 (UTC)

@Nomen4Omen: (Sorry for not answering for a while; I was occupied longer than planned with an application of subquotients.)

Ich könnte vielleich einen Nachweis finden, wenn die Aussage wahr wäre. Please, consider the following infinite example, showing that not even the subgroup relation is "antisymmetric up to isomorphisms":

$\bigoplus _{n=1}^{\infty }({\bf {Z}}/4{\bf {Z}})\ \simeq \ \bigoplus _{n=2}^{\infty }({\bf {Z}}/4{\bf {Z}})\ \subset \ (2{\bf {Z}}/4{\bf {Z}})\oplus \bigoplus _{n=2}^{\infty }({\bf {Z}}/4{\bf {Z}})\ \subset \ \bigoplus _{n=1}^{\infty }({\bf {Z}}/4{\bf {Z}})$.

For me, definitely, the biggest problem is to get clear definitions and correct statements in the first place. References and/or explicit proofs come second. Regards, JoergenB (talk) 15:51, 2 July 2024 (UTC)