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67 votes
1 answer
7k views

Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof. An earlier question, Measures of non-abelian-ness was thoroughly answered by Arturo Magidin. A paper by Gustafson1 proves that, for a nonabelian group, ...
Joseph O'Rourke's user avatar
42 votes
6 answers
4k views

Measures of non-abelian-ness

Let $G$ be a finite non-abelian group of $n$ elements. I would like a measure that intuitively captures the extent to which $G$ is non-commutative. One easy measure is a count of the non-commutative ...
Joseph O'Rourke's user avatar
23 votes
3 answers
1k views

In an inductive family of groups, does the probability that a particular word is satisfied converge?

We have some group word $w$ in $k$ letters. We say a $k$-tuple of group elements $\vec{g} = (g_1, g_2, \ldots , g_k) \in G^k$ satisfies the word $w$ if $w$ gives the identity at $\vec{g}$. More ...
John Wiltshire-Gordon's user avatar
21 votes
2 answers
1k views

Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$. If there are $g(n)$ non-isomorphic groups of order $n$, ideally each group would occur with probability $1/g(n)...
Joseph O'Rourke's user avatar
21 votes
0 answers
578 views

Density of first-order definable sets in a directed union of finite groups

This is a generalization of the following question by John Wiltshire-Gordon. Consider an inductive family of finite groups: $$ G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i \...
Gene S. Kopp's user avatar
  • 2,200
13 votes
1 answer
791 views

How nearly abelian are nilpotent groups?

It is not uncommon to read that "nilpotent groups are 'close to abelian'."1,2 Can this sentiment be made precise in the sense of the Turán and Erdős definition of "the probability that two elements of ...
Joseph O'Rourke's user avatar
12 votes
2 answers
406 views

Does asymmetric fraction of finite groups tend to $0$?

Let’s define asymmetric fraction of a finite group $G$ as the number $$\mathrm{af}(G) = \frac{|\{(g, a) \in G \times \mathrm{Aut}(G)\mid a(g) = g\}|}{|G|\cdot|\mathrm{Aut}(G)|}.$$ Equivalently it can ...
Chain Markov's user avatar
  • 2,618
7 votes
1 answer
548 views

The probability that two elements of a finite nonabelian simple group commute

It is mentioned in here (last paragraph of the first page) that Dixon proved the following result: the probability that two elements of a finite nonabelian simple group commute is at most $\frac{1}{12}...
user129021's user avatar
7 votes
0 answers
233 views

Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?

Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...
Alexander Chervov's user avatar
4 votes
0 answers
266 views

Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)

In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
Alexander Chervov's user avatar
2 votes
0 answers
202 views

Random walk on a finite group, converging modulo a function

Let $G$ be a finite group, and let $Q$ be a probability measure on $G$. Suppose that $Q$, as a function on $G$, is supported on a conjugacy class $C$. We denote by $Q^{*k}$ the $k$-fold convolution of ...
Ofir Gorodetsky's user avatar
1 vote
0 answers
489 views

Can we generalize the concept of "characters" in group theory via methods from statistics and probability theory?

$\DeclareMathOperator\Cov{Cov}$Motivation: If $G$ is a finite group and $\phi=X+iY: G\to \mathbb{T}$ is a character of $G$, then $\Cov(X,Y)=0$ where $X$, $Y$ are considered as two real random ...
Ali Taghavi's user avatar
1 vote
0 answers
311 views

Show that $\mathrm{SL}_2(\mathbb{F}_p)$ is quasi-random

Terry Tao gives this oblique definition of quasirandom group in his notes 3 $G$ is quasi-random (of order $D$) if all non-trivial unitary representations $\rho: G \to U(H)$ have dimension at least $...
john mangual's user avatar
  • 22.8k
0 votes
1 answer
153 views

Probability distribution of random products of elements of a generating set of a finite non-abelian group

Let $G$ be a finite non-abelian group, and consider a choice of $N$ distinct elements $g_{0},g_{1},\ldots,g_{N-1}\in G$ that generate $G$. Now, let $t$ be an arbitrary positive integer, and let $d_{1},...
MCS's user avatar
  • 1,284