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Given a finite, non-abelian group $G$ of order $n$, how close can $G$ be to being abelian? More formally, define the following density measure:

$$d(G) = \frac{\#\{(a, b) \mid a, b \in G, ab = ba\}}{n^2}$$

The question is, then, are there known upper bounds for $d(G)$? I suppose lower bounds might also be interesting. A naive lower bound for $d(G)$ would be $\frac{2n-1}{n^2}$, since the commutator is trivial whenever $a = b$ or $a = e_G$ -- is this tight (in whatever sense you see fit, e.g. asymptotically)?

MT_
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    As for the lower bound, you might want to look into "perfect groups" which are those whose abelianization is trivial (i.e. $ab=ba$ only if they're the same or one is the identity), which gives you exactly the bound you described. For convenience, the smallest non-trivial perfect group is $A_5$. – Sebastian Schulz Mar 28 '17 at 04:58
  • Did you mean to define $d(G)$ as the number of commuting pairs? It sounds like you did, from the second paragraph. – Chris Culter Mar 28 '17 at 05:20
  • @SebastianSchulz Can you give a source? I don't see why the commutator is equal to $G$ doesn't mean there can't exist distinct, non-identity elements for which $[a, b] = 1$. Also, on a more philosophical level, this ties into the question of what's the best way to define the notion of (upper/lower) bound in the context of this question since (for example) I assume that there aren't perfect groups of every order. – MT_ Mar 28 '17 at 05:21
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    I vaguely recall there being a theorem which says that if this fraction exceeds 3/4 then the group is abelian, and that this is the best possible upper bound for non-abelian groups. See also https://groupprops.subwiki.org/wiki/Commuting_fraction

    My guess is that this ratio converges to 0 for $S_n$ as $n$ tends to infinity, and that this is also known. There are lots of papers on commuting probabilities in symmetric groups and the types of sums that appear when expressing the fraction in terms of centralizers.

     – zibadawa timmy Mar 28 '17 at 05:39
  • Of course a perfect group has commuting elements. – Mariano Suárez-Álvarez Mar 28 '17 at 05:39
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    In a finite non abelian group, this is lower than $5/8$ : $d(G) =1$ or $d(G)\leq 5/8$. And this is the best bound, because there are examples where $d(G)=5/8$ – Maxime Ramzi Mar 28 '17 at 05:43
  • @Max That must have been what I was thinking of, then. Indeed it's in my link and I failed to notice. Still leaves open the question of the lower bound. – zibadawa timmy Mar 28 '17 at 05:48
  • @MCT Thanks for your answer, you are of course right, I messed up quotients and subgroups in the heat of the moment. A perfect group is one whose abelianization is trivial, I will delete my additional remark. Note that the commutators are somewhat described by Grün's lemma: If you take the quotient of a perfect group by its center, you'll end up with something centerless. In particular, there are perfect groups with non-trivial center (such as SL$(2,5;\mathbb{Z})$. – Sebastian Schulz Mar 28 '17 at 06:19
  • Never mind the deleting part. I wasn't aware that SE doesn't allow editing of comments after a time window closes, so now I'll look stupid for eternity ;-) – Sebastian Schulz Mar 28 '17 at 06:21
  • @zibadawatimmy thank you, I understand, however I didn't mean to entirely delete my remarks – Sebastian Schulz Mar 28 '17 at 06:31

1 Answers1

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This is known as the commuting probability. A strict upper bound is known: $d(G)\leq 5/8$ if $G$ is non-abelian, and $d(G)=1$ if $G$ is abelian. This is mentioned both on the prior link, and is available on the SE. Indeed, $$d(G)=5/8 \iff \text{Inn}(G)\cong \mathbb Z_2\times\mathbb Z_2,$$ which is implicitly proved on that aforementioned SE Q&A. Some explicit examples include the quaternion and dihedral groups of order 8.

As for lower bounds, consider this arxiv paper. The introduction says that in the late 60's it was shown (by Erdos et al.) that \begin{align} d(G) \geq \frac{\text{log}_2\, \text{log}_2 |G|}{|G|}. \end{align} It also includes several references to other results on $d(G)$, including quite a number of characterizations and classifications for groups satisfying a certain upper bound.

This paper of Keller also includes and summarizes several results on lower bounds.

Going through those papers and their references should provide you with a wealth of results.

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