The Odds 3 (or More) Group Elements Commute - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T09:58:47Z http://mathoverflow.net/feeds/question/107904 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107904/the-odds-3-or-more-group-elements-commute The Odds 3 (or More) Group Elements Commute John Mangual 2012-09-23T14:31:58Z 2012-10-01T21:46:39Z <p>Some time ago I asked about <a href="http://mathoverflow.net/questions/91685/5-8-bound-in-group-theory" rel="nofollow">the odds 2 group elements commute</a>. I wonder about the odds that 3 group elements commute. Is there a "closed" formula for the sum</p> <p>$$\frac{1}{|G|^3} \sum_{g,h,k} \delta([g,h]=1)\delta([h,k]=1)\delta([k,g]=1)$$</p> <p>One approach, as mentioned in Kefeng Liu, might be to use the "<a href="http://arxiv.org/abs/math/9910101" rel="nofollow">Heat Kernel</a>" for finite groups.</p> <p>$$H(t,x,y) = \frac{1}{|G|} \sum_{\lambda \in \mathrm{Irr}(G)} \mathrm{dim}(\lambda) \chi_\lambda(xy^{-1}) e^{-t f(\lambda)}$$</p> <p>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:</p> <p>$$\delta(xy^{-1})= \frac{1}{|G|} \sum_{\lambda \in \mathrm{Irr}(G)} \chi_\lambda(x) \overline{\chi_\lambda(y)} = \frac{1}{|G|} \sum_{\lambda \in \mathrm{Irr}(G)} \chi_\lambda(1) \chi_\lambda(xy^{-1})$$</p> http://mathoverflow.net/questions/107904/the-odds-3-or-more-group-elements-commute/107909#107909 Answer by Geoff Robinson for The Odds 3 (or More) Group Elements Commute Geoff Robinson 2012-09-23T15:48:12Z 2012-09-29T02:28:08Z <p>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 <code>$\sum_{x \in G} k(C_{G}(x))|C_{G}(x)|$</code>. If $G$ has $k$ conjugacy classes, with representatives <code>$\{ x_{i} : 1 \leq i \leq k \}$</code>, this may be rewritten as <code>$|G| \sum_{i=1}^{k} k(C_{G}(x_{i})).$</code> The probability you require, assuming a uniform distribution, is <code>$\frac{1}{|G|^{2}}\left( \sum_{i=1}^{k} k(C_{G}(x_{i})) \right)$</code>. 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 $|G|^{2}$. </p> <p>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 <code>$c_{n+1}(G) = \sum_{x \in G} c_{n}(C_{G}(x))$</code>.</p> http://mathoverflow.net/questions/107904/the-odds-3-or-more-group-elements-commute/108392#108392 Answer by François Brunault for The Odds 3 (or More) Group Elements Commute François Brunault 2012-09-29T07:30:36Z 2012-09-29T07:30:36Z <p>This question is studied in the article <a href="http://dx.doi.org/10.1006/jabr.1995.1331" rel="nofollow">Isoclinism classes and commutativity degrees of finite groups</a> 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 $Q_8$.</p>