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This tag is used if a reference is needed in a paper or textbook on a specific result.

9 votes
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Results about the existence of solutions in groups

If you are interested in finite simple groups, there is a whole raft of literature considering the related question of which words maps $w: G^n\to G$ are surjective. The answer is often yes, so this g …
Nick Gill's user avatar
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3 votes
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Conjugacy classes in PSL(3,q) and PSU(3,q)

This is well-known, and there are a number of relevant references. Firstly, there are these by Wall (they are pretty hard to read though). Wall, G. E. Conjugacy classes in projective and special line …
Nick Gill's user avatar
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11 votes
3 answers
820 views

Finite groups with few conjugacy classes of maximal subgroups

Let $c$ be a positive integer, $G$ a finite group with at most $c$ conjugacy classes of maximal subgroup. What can we say about $G$? Same question, but this time $G$ is a finite group with at most $ …
Nick Gill's user avatar
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8 votes
Accepted

Finite groups factorized into two simple alternating groups

For $m=5$ and $n$ arbitrary, all factorizations are classified here: W. R. Scott, Products of $A_5$ and a finite simple group, J. Algebra 37 (1975), 165--171. For $m=6$ and $m=7$ (again with $ …
Nick Gill's user avatar
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3 votes
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On Groups of Maximal Class: Reference

I believe the study of $p$-groups of maximal class really kicked off with this paper: Blackburn, N. On a special class of p-groups. Acta Math. 100 1958 45–92. These days the basic source is: …
Nick Gill's user avatar
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17 votes
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Reference for the triple covering of A_6

The oval construction for $3\cdot A_6$ can be found on p.110 of Symmetric Generation of Groups With Applications to Many of the Sporadic Finite Simple Groups by Robert Curtis. An e-version is …
Nick Gill's user avatar
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4 votes

Subgroups of GL_2 over a finite field

Dickson is responsible for the classification of subgroups of $SL_2(\mathbb{F}_q)$ (and once you've got this the subgroups of $GL_2(\mathbb{F}_q)$ are easy). You can find a full proof in Suzuki's "Gro …
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3 votes
0 answers
54 views

Cliques in Incomplete block designs

I'm interested in inequalities that guarantee the presence of cliques in incomplete block designs. Here's the set-up: I have an incidence structure $(V, B)$ which is an incomplete block design: $V$ is …
Nick Gill's user avatar
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7 votes

Realizable Order Sequences for Finite Groups

I don't know if this constitutes an answer but... You might be interested in the paper by Mazurov called The set of orders of elements in a finite group. Given a group $G$, Mazurov defines $\omega(G …
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5 votes

A family of skew-symmetric matrices corresponding to cycles in graphs

This isn't a proper answer, but it's slightly too long for a comment... A matrix that satisfies (i) and (ii) is biregular and skew-symmetric. In principle a biregular skew-symmetric matrix could have …
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10 votes

Measures of non-abelian-ness

For certain applications, the abelian-ness of a group is inversely proportional to the quasirandom-ness of a group. The latter notion is more obscure, so this observation might not be a help at first. …
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7 votes

Letter from Grothendieck to Tate on "crystals"

I don't know if this is of any use.... but I believe that the ideas in this letter were written up somewhat later in this article: Grothendieck, A. Crystals and the de Rham cohomology of schemes …
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7 votes
Accepted

Finite subgroups of $GL(2,K)$ with $K\neq\mathbb{C}$

An answer to your questions is provided by this article of Beauville: Beauville, Arnaud, Finite subgroups of (\mathrm{PGL}_2(K))., García-Prada, Oscar (ed.) et al., Vector bundles and complex geometr …
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2 votes
2 answers
305 views

Random walk and isoperimetric constant

I assume that a result of the following kind is known, and I would really appreciate a reference for it... Or at least, some hints as to where to start looking. Theorem(?): Let $\varepsilon>0$ and …
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9 votes

Maximal number of maximal subgroups

The document I linked to above is sufficiently striking as to warrant an answer of its own. I hope it complements the community wiki above. As mentioned above the relevant conjecture in this area is …
Nick Gill's user avatar
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