Denote the commuting probability (the probability that two randomly chosen elements commute) of a finite group $G$ by $\operatorname{cp}(G)$. By a result of Gustafson [2], $\operatorname{cp}(G)=\operatorname{k}(G)/|G|$, where $\operatorname{k}(G)$ is the number of conjugacy classes of $G$. My question is the following:



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> Does there exist a function $f:\left(0,1\right]\rightarrow\left[0,\infty\right)$ such that the following holds: "If $G$ is a finite group, $\rho\in\left(0,1\right]$, and $\operatorname{cp}(G)\geq\rho$, then $G$ has an abelian normal subgroup of index at most $f(\rho)$."?  
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> Equivalently: Does there exist no sequence $(G\_n)\_{n\geq 1}$ of finite groups such that $\operatorname{cp}(G\_n)$ is bounded away from $0$, yet the minimal index of an abelian normal subgroup in $G\_n$ converges to $\infty$ as $n\to\infty$?
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This question arose while I was thinking about different definitions of "almost-abelianity" of finite groups $G$, each of which generalizes abelianity in a quantitative way and which I briefly discuss now for the sake of motivation.



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> 1. For $I,C\in\left[0,\infty\right)$, say that $G$ is *$(I,C)$-almost abelian* if and only if $[G:\operatorname{Fit}(G)]\leq I$ and $\operatorname{cl}(\operatorname{Fit}(G))\leq C$. Here, $\operatorname{Fit}(G)$ denotes the Fitting-subgroup of $G$, and $\operatorname{cl}(H)$ the nilpotency class of a nilpotent group $H$.
> 2. For $\rho\in\left(0,1\right]$, say that $G$ is *$\rho$-almost abelian* if and only if $\operatorname{cp}(G)\geq\rho$.
> 3. For $I\in\left[0,\infty\right)$, say that $G$ is *$I$-virtually abelian* if and only if $G$ has an abelian normal subgroup of index at most $I$.
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Each of the following facts about finite groups $G$ is either not difficult to see or follows from results in the literature:


1. **Virtual abelianity implies almost-abelianity in the second sense."**
More precisely, there is a function $f:\left[0,\infty\right)\rightarrow\left[0,1\right)$ such that if $G$ is $I$-virtually abelian, then $G$ is $f(I)$-almost abelian. Actually, one can choose $f:I\mapsto I^{-2}$.
2. **"Almost-abelianity in the second sense implies almost-abelianity in the first sense."** More precisely, there are functions $f\_1,f\_2:\left(0,1\right]\rightarrow\left(0,\infty\right)$ such that if $G$ is $\rho$-almost abelian, then $G$ is $(f\_1(\rho),f\_2(\rho))$-almost abelian (the existence of $f\_1$ is by [1, Theorems 4(iii) and 8(ii)], and the existence of $f\_2$ follows from [1, Lemma 2(ii)] and the fact that $\operatorname{cp}(G)\leq 5/8<1$ for all nonabelian finite groups $G$ [2]).
3. **"Almost-abelianity in the first sense does not imply almost-abelianity in the second sense."** More precisely, even if both the index and the class of the Fitting subgroup of $G$ are bounded, $\operatorname{cp}(G)$ can be arbitrarily small.


Hence the later a property is defined in the list above, the stronger it is, and the second property is strictly stronger than the first. My question asks whether the third property is also strictly stronger than the second.



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> References:
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> [1] W. H. Gustafson, What is the probability that two group elements commute?, *Amer. Math. Monthly* **80** (1973), 1031-1034.  
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> [2] R. M. Guralnick and G. R. Robinson, On the commuting probability in finite groups, *J. Algebra* **300** (2006), 509-528.
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Finite groups with lots of conjugacy classes, but only small abelian normal subgroups?
No: there exists a sequence of finite groups with commuting probability bounded away from 0 but with no abelian (normal) subgroup of bounded index.


Fix a prime power $q$. Consider the "higher Heisenberg" group $G\_n$ of order $q^{2n+1}$ consisting of those square matrices of size $n+2$ over $\mathbf{F}\_q$ of the form
$$P(u,v,x)=\begin{pmatrix}1 & u & x\\ 0 & I\_n & v\\ 0 & 0 & 1\end{pmatrix},$$
where $u$ is a row, $v$ is a column, and $x$ a scalar. If we endow $\mathbf{F}\_q^{n^2}$ with the symplectic product $\langle u\oplus v,u'\oplus v'\rangle=uv'-u'v$, then we see that the centralizer of $P(u,v,x)$ is the set of $P(u',v',x')$ such that $\langle u\oplus v,u'\oplus v'\rangle=0$. In particular, this is a subgroup of index $q$ (unless $(u,v)=(0,0)$ in which case $P(u,v,x)$ is central). So the probability that two elements commute is $\ge 1/q$ [actually it's $1/q+q^{-2n}(1-1/q)$, if I'm correct]. 


On the other hand, the largest cardinal of an abelian subgroup in $G\_n$ is $q^{n+1}$ (since $n$ is the largest dimension and hence $q^n$ is the largest cardinal of an isotropic subspace in the symplectic space $\mathbf{F}\_q^{2n}$). So the minimal index of an abelian subgroup in $G\_n$ is $q^n$. Thus, $q$ being fixed ($q=2$ is fine) and $n$ tending to infinity, we get the required example.


