Linked Questions

64
votes
3answers
6k views

5/8 bound in group theory

The odds of two random elements of a group commuting is the number of conjugacy classes of the group $$ \frac{ \{ (g,h): ghg^{-1}h^{-1} = 1 \} }{ |G|^2} = \frac{c(G)}{|G|}$$ If this number exceeds ...
24
votes
3answers
2k views

Combinatorial Techniques for Counting Conjugacy Classes

The number of conjugacy classes in $S_n$ is given by the number of partitions of $n$. Do other families of finite groups have a highly combinatorial structure to their number of conjugacy classes? For ...
19
votes
4answers
808 views

The number of commuting m-tuples is divisible by order of group: Improvements?

The number of commuting pairs of elements in finite group G is equal to the product $k(G)*|G|$ (see MO271757 ) where $k(G)$ is the number of conjugacy classes. Thus it is is divisible by $|G|$ (the ...
8
votes
3answers
726 views

Web interface for GAP (or other computer algebra system dealing with finite groups)?

GAP is computer algebra system which allows to make calculations with finite groups. (See wikipedia link for an example). Is there web interface for it ? (I cannot google it.) Or may be some other ...
32
votes
2answers
894 views

Analogies supporting heuristic: Weyl groups = algebraic groups over field with one element?

There is well-known heuristic that Weyl groups are reductive algebraic groups over "field with one element". Probably the best known analogy supporting that heuristic is the limit $q\to1$ ...
8
votes
2answers
661 views

The Odds 3 (or More) Group Elements Commute

Some time ago I asked about the odds 2 group elements commute. I wonder about the odds that 3 group elements commute. Is there a "closed" formula for the sum $$ \frac{1}{|G|^3} \sum_{g,h,k} \delta([...
20
votes
1answer
433 views

$q$-(and other)-analogs for counting index-$n$ subgroups in terms of Homs to $S_n$?

The following formula of astonishing beauty and power (imho): $$ \sum_{n \ge 0} \frac{| \mathrm{Hom}(G,S_n) | }{n! } z^n = \exp\left( \sum_{n \ge 1} \frac{|\text{Index}~n~\text{subgroups of}~ G|}nz^...
6
votes
3answers
1k views

Conjugcy classes in GL(F_2) ? GL(F_q)

How to deduce a formula (see below) for number of conjugacy classes in GL_n(F_2) ? (More generally F_q) ? Is there some description of conjugacy classes or we just know how many of them but do not ...
4
votes
1answer
356 views

Are degrees of polynomials in Weil's zeta function equal/bounded to/by dimensions of SOME cohomologies in non-smooth or non-projective case?

[Edit] Let me make question more focused. It is about details of Weil conjectures. Rationality of zeta function does NOT require the manifold to be smooth & projective, so zeta function is a ...
9
votes
0answers
311 views

Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?

$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons. ...