Degree of commutativity of finite groups and subgroups

Question

Recently I started reading some articles about the degree of commutativity of finite groups. I have some questions:

1. In "Subgroup commutativity degrees of finite groups" Tarnauceanu proposes the following formula for calculating the degree of commutativity of subgroups of a finite group G:

$$ \mathrm{sd}(G) = \frac{\left| \left\{(H,K) \in \mathscr{L}(G)^2:HK=KH\right\}\right|}{\left|\mathscr{L}(G)\right|^2}, $$

He proves that if $G_1, G_2, \ldots , G_n$ are finite groups of coprime order, then

$$ \mathrm{sd}\left(\times_{i=1}^{n}G_{i}\right)=\prod_{i=1}^{n}\mathrm{sd}(G_i). $$

My first question is about what happens if we omit the hypothesis that $G_i$ have coprime orders, that is, if there exists some estimate for $$\mathrm{sd}\left(\times_{i=1}^{n}G_{i}\right)$$ in terms of $\mathrm{sd}(G_i)$.

2. In "Central Extensions and Commutativity Degree" Lescot proposes the following formula for calculating the degree of commutativity of a finite group $G$:

$$\mathrm{d}(G)=\frac{1}{|G|^2} \left|\left\{(x,y) \in G^2:xy=yx\right\}\right|.$$

My second question is about the order of the group $G$. Is there any theory for the case when $G$ is infinite? For example, $G$ might be a group equipped with a Haar measure. I have found no literature about this case.

Remark: Sorry for the obscurity of my question, the first question is: For $G_1,\ldots,G_n$ arbitrary groups, is there any estimate for the degree of commutativity of subgroups of $G$ = direct product of the groups $G_i$ in terms of the degree of commutativity of the groups $G_i$? My second question was partially answered. A friend showed me the following article: tmu.ac.ir/salg20/talks/Rezaei.pdf.

Answer 1

This is in answer to your second question. There is a note by Gustafson:

• MR0327901 (48 #6243) Gustafson, W. H. What is the probability that two group elements commute? Amer. Math. Monthly 80 (1973), 1031–1034.

where he proves the result Ben mentions, viz. if $G$ is a finite nonabelian group, then $d(G) \leq 5/8$. He goes on to prove that the same result for the case where $G$ is a compact, Hausdorff topological group (endowed with the Haar measure).

While $d(G)$ has received some attention over the years (I think it was first mentioned in a paper of Erdos in the late 60s and there have been sporadic papers since then) very little seems to have been said about $d(G)$ where $G$ is an infinite group until recently. The basic results (most of which are analogous to the finite case) are proved in

• MR2558527 (2010m:22003) Rezaei, Rashid; Erfanian, Ahmad(IR-MASHM) On the commutativity degree of compact groups. (English summary) Arch. Math. (Basel) 93 (2009), no. 4, 345–356.

Ben has already mentioned the nice paper of Levai and Pyber where it is proved that if $G$ is a profinite group and $d(G) > 0$, then $G$ is abelian-by-finite. This result is extended to all compact groups in a recent preprint by Hofmann and Russo. There is much more besides in this preprint, I'm still digesting it myself!

Answer 2

I'm not entirely sure what your question is, but I'll take it as an excuse to point out some references on the degree of commutativity of a finite group $G$. I take a (very) lay interest in this because I often set to undergraduates the problem of proving that if $G$ is nonabelian then $d(G) \leq 5/8$.

A very comprehensive discussion of this is may be found in this 1979 paper of Rusin:

Link

There's also a 1983 article in Eureka (the magazine of the Cambridge student maths society) by Nigel Boston called "Nearly abelian groups", which (from memory) gives a fun exposition of the same thing. I've referred to this article before, so perhaps I'll take the opportunity to wander over to the library and scan it in, since this publication is not widely available outside Cambridge.

Secondly I'd like to draw attention to a paper of Peter Neumann called "Two combinatorial problems in group theory".

MR1005821 (90f:20036) Neumann, Peter M.(4-OXQ) Two combinatorial problems in group theory. Bull. London Math. Soc. 21 (1989), no. 5, 456–458.

I chanced across it quite by accident about 10 years ago. It proves the following very nice result: if $d(G) \geq \alpha$ then there are normal subgroups $K \leq H \lhd G$ with $[G : H] \leq C_1(\alpha)$, $|K| \leq C_2(\alpha)$, and $H/K$ abelian. Roughly, the only way you can have a positive proportion of elements commuting is if $G$ is virtually (small-by-abelian).

To answer your last question, I think the following paper may be relevant.

MR1764885 (2001i:20059) Lévai, L.; Pyber, L.(H-AOS) Profinite groups with many commuting pairs or involutions. (English summary) Arch. Math. (Basel) 75 (2000), no. 1, 1–7

Update: I visited the library and scanned the Eureka article. A PDF is available here: http://www.dpmms.cam.ac.uk/~bjg23/papers/boston.pdf

Answer 3

I am quite late on this one, but here's a partial answer to your first question. First, a couple of comments:

1. I take the view that the term "commutativity" is unfortunate when applied to subgroups; permutability is more appropriate.

2. There is really no need to talk about the subgroup lattice of the group $G$. Lattices have structure, but here we only need their size, so set of subgroups, which you may denote by $\mathrm{s}(G)$ for instance, would be just as good.

The coprime orders case you mention (Proposition 2.2 of the original article) is, of course, valid. If $\gcd(|A|,|B|)=1$, then every subgroup of $A \times B$ is a subproduct, i.e., a direct product of subgroups of the two factors. This follows directly from Goursat's lemma; you can have a look at

• D.D. Anderson and V. Camillo, Subgroups of direct products of groups, ideals and subrings of direct products of rings, and Goursat’s lemma, Rings, modules and representations. International conference on rings and things in honor of Carl Faith and Barbara Osofsky, Zanesville, OH, USA, June 15–17, 2007, American Mathematical Society, 2009, pp. 1–12.

for an excellent account of this useful result (and some generalisations). As far as I can tell, Tarnauceanu does not offer any justification for Proposition 2.2 in his paper, that is why I mention this.

The following properties are all satisfied by $\mathrm{d}$, but not by $\mathrm{sd}$:

1. If $G_1$, $G_2$ are groups then $\mathrm{d}(G_1 \times G_2) = \mathrm{d}(G_1) \cdot \mathrm{d}(G_2)$.
2. If $N$ is a normal subgroup of $G$ then $\mathrm{d}(G) \leq \mathrm{d}(N) \cdot \mathrm{d}(G/N)$.
3. If $H$ is a subgroup of $G$ then $\mathrm{d}(H) \geq \mathrm{d}(G)$.

For instance, the dicyclic group of order 12 with presentation $$\mathrm{Dic}_3 = \left\langle a, x : a^{6} = 1, x^2 = a^3, x^{-1}ax = a^{-1}\right\rangle,$$ has $\mathrm{sd}\left( \mathrm{Dic}_3\right) = 29/32$ and a normal subgroup of order 2 with quotient isomorphic to $\Sigma_3$, but $\mathrm{sd}\left( \Sigma_3\right) = 5/6$, thus 2. is not true. Then the group of order 16 which is a central product of $D_8$ and $C_4$ over a common cyclic central subgroup of order 2 has subgroup permutability degree equal to $505/529$, but $\mathrm{sd}\left( D_8 \right) = 23/25$, thereby disproving 3. Finally, $\mathrm{sd}\left( C_2 \times \Sigma_3\right) = \mathrm{sd}\left(D_{12} \right) = 101/128$, but $\mathrm{sd}\left( \Sigma_3\right) = 5/6$, hence 1. does not hold either.

Answer 4

You may be interested in this paper, which answers a generalization of your question about the infinite case: http://arxiv.org/abs/1102.4353

Gene and I do not yet know how our work fits in with the larger picture.