Definition talk:Internal Group Direct Product

Mentions about restrictions of operation in subgroup should be moved from here to the definition of the subgroup for grater clarity as for they don't contribute to general understanding anyway. joel ^talk 17:27, 29 December 2012 (UTC)

Such restriction is already discussed in full in Definition:Restriction/Operation. In this instance I disagree with you: knowing exactly to which subset the given operation is restricted is useful for understanding exactly what is going on here. --prime mover (talk) 17:33, 29 December 2012 (UTC)

But this is clear from the definition of the subgroup—you're doubling it here! Please kindly reconsider your view on this matter—those restrictions don't bring anything new: if you'd like to leave them as they are then you could resign from the word "subgroup" there and say something like "Let $H$ be a subset of a group $\left({G, \circ}\right)$. Taking restriction $\circ \restriction_H$ of $\circ$ to $H$ makes $H$ a group with this operation, i.e. $\left({H, \circ \restriction_H}\right)$ is a group." This group is a subgroup of $\left({G, \circ}\right)$. But this is pointed out in the article about subgroup definition, not explicitly and maybe herein lies the problem. joel ^talk 19:06, 29 December 2012 (UTC)

Reconsidering my view ... there. Reconsidered. Leaving it as it is. --prime mover (talk) 21:21, 29 December 2012 (UTC)

According to changes made recently to definition of subgroup we could safely resign here from talking about restrictions… it seems that everything is clear here too. joel ^talk 21:18, 3 January 2013 (UTC)

Despite your suggestion, it is better to leave it as it was, with full detail about the restrictions. --prime mover (talk) 22:54, 9 January 2013 (UTC)

BTW please don't refer to such amendments as "corrections" unless they replace something which is genuinely incorrect. --prime mover (talk) 22:55, 9 January 2013 (UTC)

By the way: most general definition is one using general indexing set for arbitrary indexed family of groups (one should distinguish group direct product and group [cartesian/general] product). joel ^talk 17:31, 29 December 2012 (UTC)

As for the definition using the general indexing set, that can be considered as work in progress. No reference to such an object has been encountered in my own reading (and I'm the only contributor in this area) - all internal direct products I've met have been finite (and therefore covered by the existing definition). --prime mover (talk) 17:35, 29 December 2012 (UTC)

I found quite interesting book of László Fuchs (Infinite abelian groups in two volumes) where I encountered both of them. I can try to find it for you on the Internet once again (I have an e-copy of it). This would mean that you'd have to dig WP trying to find instances of direct product and cartesian product of groups (those are often named "direct sum" and "direct product" which is misleading to me: your assistance would be very profitable) and differentiate them; I forgot to mention explicitly that when indexing sets are finite then both notions coincide. joel ^talk 19:06, 29 December 2012 (UTC)

Feel free to post up an impression of how you envisage such a definition somewhere in your user domain. We'll be glad to check it out! --Lord_Farin (talk) 18:58, 29 December 2012 (UTC)

Sure, I'll try! I mean I probably rewrite it from the aforementioned book! :p joel ^talk 19:06, 29 December 2012 (UTC)

Why bother with a theorem?

Why do we have both https://proofwiki.org/wiki/Internal_Direct_Product_Theorem and https://proofwiki.org/wiki/Definition:Internal_Group_Direct_Product ? Why can't we just have a 3rd definition? BCLC (talk) 01:13, 28 October 2021 (UTC)

I'm more and more wondering if the question really ought to be "Why bother with $\mathsf{Pr} \infty \mathsf{fWiki}$?" Less and less does it look fit for purpose. --prime mover (talk) 05:09, 28 October 2021 (UTC)