## The Odds 3 (or More) Group Elements Commute

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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

$$\frac{1}{|G|^3} \sum_{g,h,k} \delta([g,h]=1)\delta([h,k]=1)\delta([k,g]=1)$$

One approach, as mentioned in Kefeng Liu, might be to use the "Heat Kernel" for finite groups.

$$H(t,x,y) = \frac{1}{|G|} \sum_{\lambda \in \mathrm{Irr}(G)} \mathrm{dim}(\lambda) \chi_\lambda(xy^{-1}) e^{-t f(\lambda)}$$

If I'm not mistaken $f(\lambda)$ is the quadratic Casimir, but not sure. Really, for $t=0$ it reduces to the group theory identity:

$$\delta(xy^{-1})= \frac{1}{|G|} \sum_{\lambda \in \mathrm{Irr}(G)} \chi_\lambda(x) \overline{\chi_\lambda(y)} = \frac{1}{|G|} \sum_{\lambda \in \mathrm{Irr}(G)} \chi_\lambda(1) \chi_\lambda(xy^{-1})$$

-

Could you fix the arXiv link to go to the abstract rather than directly to the PDF? Thank you! – Harry Altman Sep 29 at 3:25

## 2 Answers

There are $k(G)|G|$ commuting pairs $(y,z)$ of elements of $G.$ How many commuting triples $(x,y,z)$ of elements of $G$ are there? If we fix the first component $x$ of the triple, note that $x$ permutes (by conjugation) the commuting pairs of elements of $G,$ and the only such pairs it fixes are the commuting pairs which already have both components in $C_{G}(x).$ Hence $x$ fixes $k(C_{G}(x))|C_{G}(x)|$ commuting pairs, and the total number of commuting triples in $G \times G \times G$ is `$\sum_{x \in G} k(C_{G}(x))|C_{G}(x)|$`. If $G$ has $k$ conjugacy classes, with representatives `$\{ x_{i} : 1 \leq i \leq k \}$`, this may be rewritten as `$|G| \sum_{i=1}^{k} k(C_{G}(x_{i})).$` The probability you require, assuming a uniform distribution, is `$\frac{1}{|G|^{2}}\left( \sum_{i=1}^{k} k(C_{G}(x_{i})) \right)$`. The discussion (and Burnside's counting lemma) makes it clear that this is (number of orbits of $G$ by conjugation on commuting pairs of elements of $G$), divided by $|G|^{2}$.

Later edit: Since I noticed the "(or more)" in the question title, the pattern is now clear: let $c_{n}(G)$ denote the number of commuting (ordered) $n$-tuples of elements of $G$. Then we have `$c_{n+1}(G) = \sum_{x \in G} c_{n}(C_{G}(x))$`.

-

5
For symmetric groups there is the beautiful identity $$\prod_{j=1}^\infty (1-u^j)^{-\sigma(j)} = \sum_{n=1}^\infty \frac{T(n)}{n!} u^n$$ where $T(n)$ is the number of triples of commuting elements in the symmetric group $S_n$ and $\sigma$ is the sum of divisors function. See arxiv.org/abs/1203.5079 by John Britnell for an elementary proof. – Mark Wildon Sep 23 at 18:10

1
For the generalization to commuting $k$-tuples of elements in $S_n$ and related results, see Exercise 5.13 in Enumerative Combinatorics, vol. 2. – Richard Stanley Oct 2 at 0:38

### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.

This question is studied in the article Isoclinism classes and commutativity degrees of finite groups by P. Lescot. The probability that $n+1$ elements of $G$ pairwise commute is called there the $n$-th commutativity degree $d_n(G)$. A kind of recursive formula for $d_n(G)$ in terms of $d_{n-1}$ of centralizers is proved (Lemma 4.1). Lescot also proves that if $G$ is not abelian then $d_n(G) \leq \frac{3 \cdot 2^n - 1}{2^{2n+1}}$ with equality if and only if $G$ is isoclinic to the quaternion group $Q_8$.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 37, "mathjax_display_tex": 4, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8956416249275208, "perplexity_flag": "head"}