Talk:Group extension

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I propose this article be moved to group extension, which already redirects here and doesn't require disambiguation. Deco 17:42, 9 December 2005 (UTC)Reply

I think some disambiguation is nesessary, but only as a redirect. That doesn't change the fact that this article is a stub; I believe it should be merged into the main topic Algebraic extension. I know that an extension and an algebraic extension are different things, but it would help those reading algebraic extension, if they knew what an extension was first, and simply linking it at the end of this piddly little thing is more of a "Would you like to know more?" than a "See also." IMHO. Sim 01:53, 1 April 2006 (UTC)Reply

I guess I think it's true that extension (algebra) is the wrong title for this article, group extension is the right title. I will now move it. On the other hand, I don't think it should be merged; group extensions are important and will one day grow into their own article, delving into calculations with group cohomology and such. -lethe ^{talk +} 07:45, 1 April 2006 (UTC)Reply

Very well. Withdrawing the merge proposal. Sim 19:06, 29 April 2006 (UTC)Reply

I made the article a little less of a stub DKleinecke 22:11, 16 November 2006 (UTC)Reply

The two definitions are now in conflict about which group is G and which is H. I think the second way is better and this first should be changed. What does anybody else think? DKleinecke 23:02, 21 November 2006 (UTC)Reply

What do you mean by the "second way"? The reason I exchanged G and H was that the previous version was incorrect under any possible interpretation. There still remains the question of whether G' is an extension of G by H, or of H by G. I did some searching, and concluded that the way it is on the page now is the most common usage: but one reference said that the other is sometimes used. Vegasprof 21:51, 23 November 2006 (UTC)Reply

I added a sentence about classifying extensions of one abelian group by another, as this is a particularly important special case.

Also, it seems to me that an extension G of Q should be equipped with a map to Q, not just that it has some quotient that is isomorphic to Q. Certainly in commutative algebra, an extension of Q by N refers to the whole exact sequence -- is this not the way that group theorists see it? QBobWatson 19:35, 3 October 2007 (UTC)Reply

The whole use of terms such as subquotient, semiproduct and such, can be a bit confusing to the novice. The idea that from division, the dividend and the quotient can construct the modulo remainder leads to a singular option, while given the dividend and the modulo, many possible quotients could be inferred. This implies that the group order is not commutative as integer factors, and some representation respecting this could be useful. —Preceding unsigned comment added by 217.171.129.71 (talk) 18:49, 18 June 2010 (UTC)Reply

There seems to be some controversy (and edits) as to whether ${\displaystyle E}$ in the exact sequence ${\displaystyle 1\to N\to E\to G\to 1}$ should be called an extension of ${\displaystyle G}$ by ${\displaystyle N}$ or an extension of ${\displaystyle N}$ by ${\displaystyle G}$. The former (and current version of the article) makes more sense to me, but I am surprised to see that Rotman's "Introduction to Homological Algebra" uses the latter language. There is a "warning" in the article that seems to be about this ambiguity, but its notation is now inconsistent with the current form of the article. Any thoughts or opinions? Mike Stone (talk) 14:48, 19 July 2011 (UTC)Reply

Both conventions are common in the literature, and the article needs to make this clear. If possible, the article should avoid adopting either convention, otherwise people will keep messing it up trying to "correct" it. Some authors use terminology such as "extension of a group K by a normal subgroup N" or "extension of a normal subgroup N by a group K", which makes it clear which convention is being used. If we do need to adopt a convention, I suggest wording things this way as far as possible.

One thing I've noticed though is that group properties such as "free-by-abelian" always seem to be understood according to the second convention. E.g., "G is free-by-abelian" means "G has a free normal subgroup N such that G/N is abelian", even if the author would say that this is an extension of an abelian group by a free [normal sub]group. --Zundark (talk) 16:13, 19 July 2011 (UTC)Reply

I think, this article should include the general construction of group extensions as described in “Über die Erweiterung von Gruppen I”, Otto Schreier, Monatsh. für Mathematik und Pysik, XXXIV. Band, p. 166–180 (1926) — Preceding unsigned comment added by 141.30.71.211 (talk) 15:05, 10 April 2014 (UTC)Reply

I think a better reference than Morandi's note would be "Maclane - Homology, 1975 edition, p. 124-129" and in particular Theorem 8.8. — Preceding unsigned comment added by 128.189.137.245 (talk) 00:16, 31 March 2019 (UTC)Reply

The current version of this article starts so:

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence

${\displaystyle 1\to N\to G\to Q\to 1.}$

If G is an extension of Q by N, then G is a group, N is a normal subgroup of G and the quotient group G/N is isomorphic to the group Q.

This is a bit strange when we regard the following example.

Let be ${\displaystyle N=(\mathbb {Z} ,+)}$, ${\displaystyle G=(\mathbb {Z} ,+)}$, ${\displaystyle Q=\mathbb {Z} /3\mathbb {Z} }$ Consider the following short exact sequuence:

${\displaystyle \{0\}\to \mathbb {Z} \;{\overset {\iota }{\to }}\;\mathbb {Z} \;{\overset {\pi }{\to }}\;\mathbb {Z} /3\mathbb {Z} \to \{0\}}$

where ${\displaystyle \iota :\mathbb {Z} \to \mathbb {Z} :z\mapsto 3z}$ and ${\displaystyle \pi }$ maps each integer to its modulo-3 equivalence class. Then according to the quotation above, we should say that

${\displaystyle \mathbb {Z} }$ is a normal subgroup of ${\displaystyle \mathbb {Z} }$ and the quotient group ${\displaystyle \mathbb {Z} /\mathbb {Z} }$ is isomorphic to the group ${\displaystyle \mathbb {Z} /3\mathbb {Z} }$.

which is not true, because ${\displaystyle \mathbb {Z} /\mathbb {Z} }$ has only one element while ${\displaystyle \mathbb {Z} /3\mathbb {Z} }$ has 3.

This can be corrected so:

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence

${\displaystyle 1\to N\;{\overset {\iota }{\to }}\;G\;{\overset {\pi }{\to }}\;Q\to 1}$.

If G is an extension of Q by N, then G is a group, ${\displaystyle \iota (N)}$ is a normal subgroup of G and the quotient group ${\displaystyle G/\iota (N)}$ is isomorphic to the group Q.

But I am not brave enough to perform this modification in the article. — Preceding unsigned comment added by 89.135.79.17 (talk) 06:42, 11 November 2019 (UTC)Reply

Since nobody reacted, I've performed this modification in the article. 89.135.79.17 (talk) 08:40, 12 December 2019 (UTC)Reply

I wonder whether shortly later the same should be applied. Though named factor group at that point (which is actually the same and should be unified), instead of ${\displaystyle G/N}$ I would expect ${\displaystyle G/\iota (N)}$ instead, as ${\displaystyle \iota (N)}$ and not ${\displaystyle N}$ is a subgroup of ${\displaystyle G}$.Stefan Groote (talk) 04:33, 13 September 2024 (UTC)Reply

I have corrected this now by myself. Stefan Groote (talk) 11:46, 20 November 2024 (UTC)Reply

My example of a splitting short sequence resulting in a semidirect product has been removed by an anonymous user, stating that "Every split central extension is a direct product." I see at least two counter statements to this: the article https://math.stackexchange.com/questions/339731/what-is-the-relation-between-semidirect-products-extensions-and-split-extension" entitled "What is the relation between semidirect products, extensions, and split extensions?" in the StackExchange forum, and a chapter https://www.hiroleetanaka.com/pdfs/2014-fall-122-lecture-16-semidirect.pdf entitled "Semidirect products are split short exact sequences" by Hiro Lee Tanaka. Is there anybody to decide this issue? Stefan Groote (talk) 11:21, 19 November 2024 (UTC)Reply

I think it would be nice to have a section of properties of groups which are closed under taking extensions – for example, an extension of a noetherian group by a noetherian group is noetherian. Does anyone else agree that this would be good to include? Joel Brennan (talk) 20:08, 1 April 2022 (UTC)Reply

• Yes, this would probably be useful. In the Quotient group article the analogous information is given as a single sentence in the Properties section. But there's currently no Properties section at all in this article, so a separate section makes sense. --Zundark (talk) 08:32, 2 April 2022 (UTC)Reply

The article mentions that the problem is “very hard”, but it doesn’t say what that means. Is it an undecidable problem? If so, this would imply that no algorithm can exist for enumerating all finite groups; the set of all finite groups would be uncomputable. This is not the case for finite _simple_ groups (since they have been completely classified and comprise a small number of infinite families plus some sporadic cases). But such a classification might not be necessarily possible for finite groups in general. 2604:2D80:6984:3800:0:0:0:77FB (talk) 01:43, 10 October 2023 (UTC)Reply

An algorithm for what problem? What are you talking about? As far as I can see, the only thing described in the article as a “problem” is the “extension problem”, which is not an algorithmic decision problem, but a loose research program of trying to come up with a satisfactory classification of properties of possible extensions (it uses the word “problem” in its ordinary English meaning, not as any technical term).

The set of all finite groups is obviously computable, and it is quite trivial to write down an algorithm that lists all extensions of G by N for given groups G and N, if that’s what you have in mind. But this is not the “extension problem” as described in the article.—Emil J. 13:17, 6 May 2026 (UTC)Reply

The redirect Trivial extension has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 December 24 § Trivial extension until a consensus is reached. Jay 💬 11:31, 24 December 2023 (UTC)Reply