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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
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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
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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
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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
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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?
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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
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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.
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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.
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use the "integral homology" perspective and feel some sort of anxiety at having Schur's "projective representations" perspective presented. Because of our core policy,
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The second paragraph and the first need to be directed to later where the identification of the cohomology and
Homology is made. John McKay
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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.
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I think the old and recent work on exponents of schur multipliers are pretty cool, consider adding or linking to it.
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Cite the recent work on efficient presentations and check if they are done. Easy search on mathscinet.
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But since the phrase "algebraic dual group" is not clear, the meaning of this passage is not clear.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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1264:{\displaystyle H_{2}(G,\mathbb {Z} )\cong {\bigl (}H^{2}(G,\mathbb {C} ^{\times }){\bigr )}^{*}}
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Emphasize the impact of the Schur multiplier more in the section on efficient presentations
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when it is viewed as a topological group? If so, then this should be stated explicitly.)
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Schur multiplier places an inequality on two fundamental invariants of a presentation."
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More precisely define the Schur cover (it has two conditions, only one given)
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Many thanks! That is exactly what I wanted. I must learn to use GAP myself.
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More carefully define terms in the section on efficient presentations too
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These were just created from GAP's
AbelianInvariantsMultiplier command.
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I hope someone knowledgeable on the subject can fix this.
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Cite the 1904, 1907 articles of Schur, the 1942 of Hopf
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