Talk:Schur multiplier

TODO

Cite the 1904, 1907 articles of Schur, the 1942 of Hopf
Mention the projective representations stuff works with only the obvious change over any algebraically closed (or solubly closed even) field. Changes are most 'obvious' for algebraic completions of finite fields, so probably stick with that.
Cite the recent work on efficient presentations and check if they are done. Easy search on mathscinet.
I think the old and recent work on exponents of schur multipliers are pretty cool, consider adding or linking to it.
There is a ton of work on multipliers of p-groups not even touched on here. It's even mentioned in Berkovich's character theory texts I think. Probably ought to go find out why.
More precisely define the Schur cover (it has two conditions, only one given)
Emphasize the impact of the Schur multiplier more in the section on efficient presentations
More carefully define terms in the section on efficient presentations too

JackSchmidt 19:24, 9 July 2007 (UTC)[reply]

Misplaced comments

The second paragraph and the first need to be directed to later where the identification of the cohomology and Homology is made. John McKay 130.54.16.90 (talk) 10:00, 6 June 2008 (UTC)[reply]

As near as I can tell, you are suggesting that the cohomological applications follow the homological applications. The article is written in a roughly chronological method, beginning with Schur's work on cohomology. The discovery of the homological interpretation is secondary, and is mentioned at the correct time. JackSchmidt (talk) 15:39, 6 June 2008 (UTC)[reply]

Relation to Efficient Presentations: 'In many areas of mathematics groups almost always originate from a presentation' - This is not mathematical writing! Needs Omission or re-writing with content. John McKay130.54.16.90 (talk) 10:05, 6 June 2008 (UTC)[reply]

This is an encyclopedia, not a mathematics journal article or book. This is called an opening sentence, and it establishes context for the lay reader. "To many people, presentations are of primary importance. Therefore the study of presentations is interesting and is called such and such. The Schur multiplier places an inequality on two fundamental invariants of a presentation."

One of the major themes of this article is that different people know the Schur multiplier through different perspectives. I have rarely discussed the Schur multiplier with someone who effortlessly considers all perspectives simultaneously. If I read your earlier comment correctly, you appear to use the "integral homology" perspective and feel some sort of anxiety at having Schur's "projective representations" perspective presented. Because of our core policy, WP:NPOV, all major views must be represented in a balanced manner. JackSchmidt (talk) 15:39, 6 June 2008 (UTC)[reply]

More examples?

Would it be reasonable to ask for more examples, maybe a list of the Schur multipliers of all groups with fewer than 32 elements? This would mean, of all groups of orders 4, 8, 9, 12, 16, 18, 20, 24, 28? Maproom (talk) 20:30, 6 January 2011 (UTC)[reply]

One would need a source for such things. Here is a little table if you want to just see what sort of information might be interesting. I skip those groups all of whose Sylow subgroups are cyclic.

Order	ID	Name	Multiplier
4	2	2×2	2
8	2	4×2	2
8	3	D₈	2
8	4	Q₈	1
8	5	2×2×2	2×2×2
9	2	3×3	3
12	3	A₄	2
12	4	D₁₂	2
12	5	3×2×2	2

Order	ID	Name	Multiplier
16	2	4×4	4
16	3	2⋉(4×2)	2×2
16	4	4⋉4	2
16	5	8×2	2
16	6	2⋉8	1
16	7	D₁₆	2
16	8	QD₁₆	1
16	9	Q₁₆	1
16	10	4×2×2	2×2×2

Order	ID	Name	Multiplier
16	11	2×D₈	2×2×2
16	12	2×Q₈	2×2
16	13	2⋉(4×2)	2×2
16	14	2×2×2×2	2×2×2×2×2×2
18	3	3×S₃	1
18	4	2⋉(3×3)	3
18	5	2×3×3	3
20	4	D₂₀	2
20	5	5×2×2	2

Order	ID	Name	Multiplier
24	3	SL(2,3)	1
24	4	Q₈⋉3	1
24	5	4×S₃	2
24	6	D₂₄	2
24	7	2×(4⋉3)	2
24	8	2⋉(2×2×3)	2
24	9	4×3×2	2
24	10	3×D₈	2
24	11	3×Q₈	1

Order	ID	Name	Multiplier
24	12	S₄	2
24	13	2×A₄	2
24	14	2×2×S₃	2×2×2
24	15	2×2×2×3	2×2×2
25	2	5×5	5
27	2	9×3	3
27	3	3⋉(3×3)	3×3
27	4	3⋉9	1
27	5	3×3×3	3×3×3
28	3	D₂₈	2
28	4	7×2×2	2

These were just created from GAP's AbelianInvariantsMultiplier command. JackSchmidt (talk) 01:59, 11 January 2011 (UTC)[reply]

Many thanks! That is exactly what I wanted. I must learn to use GAP myself.

I feel it would be useful to put this information in the article, but I don't know the best way to do so. I'll leave it to you. Maproom (talk) 14:18, 11 January 2011 (UTC)[reply]

Unclear statement

The section Relation to topology contains this sentence:

"In general,

H_{2}(G,\mathbb {Z} )\cong {\bigl (}H^{2}(G,\mathbb {C} ^{\times }){\bigr )}^{*}

where the star denotes the algebraic dual group."

But since the phrase "algebraic dual group" is not clear, the meaning of this passage is not clear.

I hope someone knowledgeable on the subject can fix this.

(Does it mean the Pontrjagin dual of $H^{2}(G,\mathbb {C} ^{\times })$ when it is viewed as a topological group? If so, then this should be stated explicitly.)50.234.60.130 (talk) 07:37, 6 December 2020 (UTC)[reply]