Talk:Quaternion group

mistake?

This needs to be fixed:

The quaternion group has presentation ({x,y} | x⁴ = y² = 1,

AxelBoldt 01:40 Dec 1, 2002 (UTC)

That's what i get for editing with the flu! :) Chas zzz brown 07:47 Dec 2, 2002 (UTC)

link to clifford algebra

on should mention that the quaternions form 2 Clifford algebras: CL(0,3)

with 1 as the scalar i,j,k as basic vectors ij jk en ki as bivectors ijk = -1 as pseudoscalar

and also the universal Clifford algebra Cl(0,2)

with 1 as scalar i and j as basic vectors k as bivector-pseudoscalar

Non-abelian subgroups?

The statement that every subgroup of Q is non-abelian doesn't make much sense- the group <1, i, -1, -i;*> is abelian. Scythe33 18:04, 26 September 2005 (UTC)

the article does not say this. By contrast, it states that every subgroup is a normal subgroup (a property shared with abelian groups), in addition every proper subgroup happens to be abelian (cyclic of order 4 or 2). Thus the quaternion subgroup is a kind of "minimal" non-abelian group (with respect to inclusion). In general, in a finite p-group of order pⁿ the (maximal) subgroups of order p^n-1 are always normal (see e.g. Frattini subgroup), but not the subgroups of smaller order p^n-j, j ≥ 2.

--212.18.24.11 11:43, 28 September 2005 (UTC)

The article does say this:

A group is called a generalized quaternion group if it has a presentation

$\langle x,y\mid x^{2^{n-1}}=1,x^{2^{n-2}}=y^{2},yxy^{-1}=x^{-1}\rangle$

for some integer n ≥ 3. The order of this group is 2ⁿ. The ordinary quaternion group corresponds to the case n = 3. The generalized quaternion groups are members of the still larger family of dicyclic groups. The generalized quaternion groups have the property that every abelian subgroup is cyclic.

This needs to be fixed. —The preceding unsigned comment was added by 137.82.36.10 (talk) 18:45, 8 May 2007 (UTC).

The article has a footnote saying that the generalized quaternion group is known by some authors as the dicyclic group. I don't know who these authors are, but Wikipedia itself discusses the same groups in two places (here and the page on dicylic groups) and only an obscure footnote and this discussion admits they are the same. This is very confusing and I spent ages trying to work out what was going on before I noticed the footnote and this discussion. Shouldn't we just replace the section on generalized quaternion groups with something about the groups with the above presentation.Jim6561 (talk) 15:00, 30 January 2010 (UTC)

Matrices

The Quaternion group can be represented as a sub-group of GL₂(С). If there are no objections to this I am going to add it to the article. along with the matrix interpretation of 1,i,j,and k--Cronholm144 10:11, 24 May 2007 (UTC)

ISBN

The ISBN number of Artin's book is wrong - it should be 978-0130047632 according to Amazon, but I don't see how to change it. (The given one corresponds to a book on ODEs.) 62.254.8.206 (talk) 10:01, 3 February 2008 (UTC)

I've fixed it. --Zundark (talk) 10:38, 3 February 2008 (UTC)

lede

After making some smaller tweaks, I reorganized the lede significantly. I removed some of the equations which were redundant (like the i^2 = j^2 = k^2 = -1, which are in the representation). I am toying with the idea of moving the "cross product" multiplications into the Properties section after the table of contents, and would appreciate any feedback. Baccyak4H (Yak!) 17:19, 6 July 2009 (UTC)

generalized quaternion group

The section generalized quaternion group fails to specify that n must be a power of 2. Most sources, e.g. Groupprops, sagemath, specify that the order of a generalized quaternion group must be a power of two. Without this restriction, a generalized quaternion group is (I think) the same as a dicyclic group; indeed, dicyclic group says "... when n is a power of 2, the dicyclic group is isomorphic to the generalized quaternion group."

I will alter the definition to specify "power of 2" unless someone points out that I am mistaken. Maproom (talk) 12:30, 23 June 2011 (UTC)

The footnotes that are already present in the article indicate that you are mistaken, in that they mention that some authors use the term as you do, but others use it as it currently is in the article. RobHar (talk) 13:03, 23 June 2011 (UTC)

So, am I right in thinking that for those others (and the writers of this article, but not of the dicyclic group article), "generalized quaternion group" means the same as "dicyclic group"? There is confusion here, I think not just in my own mind. I would like first to sort out my own understanding, and then (if necessary) to make the article a bit clearer. Maproom (talk) 13:40, 23 June 2011 (UTC)

Theorem 1 of page 45 of Johnson's book (as referenced in the article) states that the generalized quaternion group defined in this article has the same presentation as the dicyclic group defined in the dicyclic group article, so I think the answer to your question is yes (I say "think" because I am not actually familiar with any of these terms, I just dug around for references for this article 2 years ago). It appears that 2 years ago I mentioned the discrepancy between the articles in question on the talk page of dicyclic group, but no one has said anything. Perhaps now is a good time to figure out what to do. Certainly, dicyclic group needs additional references and a note concerning the different nomenclature, then it's a matter of figuring out whether to merge anything. RobHar (talk) 21:09, 23 June 2011 (UTC)

I think including dicyclic groups on the quaternion group page is a little too much. I think generalized quaternion 2-groups should be mentioned here (as very important generalizations with most of the properties of Q8), and on the dicyclic page (with very special properties not generally enjoyed by dicyclic groups). However, for now I just made the link to dicyclic groups more prominent. If I collect more results on (generalized) quaternion 2-groups, I'll add them to the article. Quaternion 2-groups are really pretty different for general dicyclic groups, but it is hard to tell that from the articles as they are now. JackSchmidt (talk) 03:05, 24 June 2011 (UTC)

Now I have two more terms to understand: "quaternion 2-groups" and "generalized quaternion 2-groups". A more general thought: if I read a textbook, its author may use terminology that is different from everyone else's, but at least he uses it consistently. If I read Wikipedia, and follow an internal link, I am quite likely to find an article which uses the same terms with different meanings. I applaud those editors who try to tidy this up. Maproom (talk) 08:21, 24 June 2011 (UTC)

Amongst finite group theorists: Q8 is called the quaternion group of order 8. A quaternion 2-group and a generalized quaternion 2-group are 2-groups (groups of order 2^n) with the presentation as given in the article. The extra "generalized" is only used to emphasize that one is speaking of more than just Q8. Most of the uses of quaternion groups are to describe Sylow subgroups or other large 2-subgroups. JackSchmidt (talk) 12:51, 24 June 2011 (UTC)

Quaternions

The article contains the statement "In abstract algebra, one can construct a real four-dimensional vector space with basis {1, i, j, k} and turn it into an associative algebra by using the above multiplication table and distributivity. The result is a skew field called the quaternions." This seems like a non-sequitur, since the above multiplication table contains group elements that are generated from the additional element −1. I've updated the notation, and now it should be more obvious that e.(1) ≠ e.(–1): one has to take the quotient by the equivalence class {1⋅e, −1⋅e}. I'll replace the statement. —Quondum 05:07, 18 October 2016 (UTC)

Dedekind & Witt

The following was recently contributed:

As R. Dedekind showed in 1887, the quaternion group can be the Galois group of an extension of the field Q of rational numbers. Reference : R. Dedekind, Ges. math. Werke II, 376-384. In 1936, E. Witt gave a simple criterion for a biquadratic extension of a field to be embeddable in a quaternionic one. Reference E. Witt, Journal für die reine und angewandte Mathematik, 174 (1936),237-245.

These references may be confirmed and adapted here. So far Dedekind has not been confirmed. They may be added to our reference, not replace it! Rgdboer (talk) 00:43, 1 September 2023 (UTC)

The collected works of Dedekind includes "Konstruktion von Quaternionkörpern" from Gottinger Digitalisierungszentrum. The construction given does not refer to a Galois group of a polynomial. The article is from Nachlass and concludes with a commentary by Wilhelm Eduard Weber. Rgdboer (talk) 21:46, 1 September 2023 (UTC)

Ernst Witt (1936) "Konstruktion von galoisschen Körpern der Characterristik p zu vorgegebener Gruppe der Ordnung p' " includes the mentioned Satz on page 244. Dedekind is cited at the conclusion. Rgdboer (talk) 22:28, 1 September 2023 (UTC)