degree of commutativity

Question

What is the exact definition of the degree of commutativity of a $p$-group? When we use notations $d(G)$ and $c(G)$ for other concepts, what is the best notation for degree of commutativity of $G$?

Answer 1

In general, the degree of commutativity is defined as the chance that two random elements commute:

$d(G)$=$\Large\frac{\#\{(x,y) \in G \times G: xy=yx\}}{|G|^2}$, also by some authors denoted as $Pr(G)$. It can be easily proved that this number equals

$\Large\frac{k(G)}{|G|}$, where $k(G)$ is the number of conjugacy classes of $G$. In general $d(G) \leq \frac{5}{8}$, if $G$ is non-commutative. And if $G$ is a non-commutative $p$-group, then

$d(G)\leq\Large\frac{p^2+p-1}{p^3}$. There is vast literature on this, see for example the Technical Report 2010-4 here.

Answer 2

Probably the definition meant in the question is that given by Nicky Hekster. However the degree of commutativity may take a completely different sense in the coclass theory of $p $ groups.

Given a finite $p$-group $G$, let $G_i$ denote its $i$th term of the lower central series.

One knows from the three subgroup lemma that $[G_i,G_j] \leq G_{i+j}$, but it may happen that $[G_i,G_j]$ is much smaller and lies in some $G_k$, with $k>i+j$, for all $i,j \geq m$, for some positive integer $m$. The degree of commutativity (with respect to $m$, which may be fixed if one fixes the coclass of $G$) $l=l(G)$ of $G$ can be defined to be the largest integer such that $[G_i,G_j] \leq G_{i+j+l}$, for all $i,j \geq m$.
Such an ingredient (in fact a slight variation of it) plays a central role in the theory of groups of maximal class.