Pyber: The Number of Pairwise Non-Commuting Elements and the Index of the Centre

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corollary_p287: Pyber's answer to Erdős's 1975 question, that a group with at most n pairwise non-commuting elements is covered by at most c^n sets of pairwise commuting elements.

example_p288: The example, credited by the paper to Isaacs, that an extraspecial group of order 2^(2m+1) has n(S) = 2m+1, centre of index 2^(2m) and cc(Gamma(S)) at least 2^m+1, so that the paper's exponential bounds are optimal in a sense.

lemma_3_1: Pyber's lemma that in a finite group with n(G) = n the largest conjugacy class has at most 4n^2 elements, the first step of the proof of the main theorem.

theorem_6_1: Pyber's main theorem that a group with at most n pairwise non-commuting elements has centre of index at most c^n, in the explicit form |G:Z(G)| <= 2^(2^25 n) 2^(3(2+2 log n)^5) of Theorem 6.1.

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L. Pyber, The number of pairwise non-commuting elements and the index of the centre in a finite group. Journal of the London Mathematical Society (2) 35 (1987), 287-295. doi:10.1112/jlms/s2-35.2.287.

For a group $G$, $n(G)$ is the largest number of pairwise non-commuting elements of $G$, and $cc(\Gamma)$ is the least number of complete subgraphs of the commuting graph $\Gamma(G)$ needed to cover $G$. The paper records Erdős's 1975 question about the maximum of $cc(\Gamma)$ in terms of $n=n(G)$, and B. H. Neumann's bound $|G:Z(G)|\le a^{n^2}$, for which Neumann asked for an improvement.

Pyber's main Theorem (p. 287) gives $|G:Z(G)|\le c^n$ for some constant $c$; Theorem 6.1 (p. 294) is its explicit form, $|G:Z(G)|\le2^{2^{25}n}2^{3(2+2\log n)^5}$ with $\log$ to base $2$. Since the cosets of the centre are abelian subsets, the Corollary (p. 287) gives $cc(\Gamma)\le c^n$. The Example (p. 288), which the paper credits to Isaacs (citing its reference [1], E. A. Bertram, Discrete Math. 44 (1983)), takes $S$ extraspecial of order $2^{2m+1}$: then $n(S)=2m+1$, $|S:Z(S)|=2^{2m}$ and $cc(\Gamma(S))\ge2^m+1$, so the paper calls both its results optimal in a sense.

The proof starts from Lemma 3.1 (p. 288), $k(G)\le4n^2$ for the largest conjugacy class size $k(G)$. With the P. M. Neumann and Vaughan-Lee bound $|G'|\le k^{\frac12(3+5\log k)}$ (Lemma 3.2, p. 289), Lemma 3.3 (p. 289) gives a subgroup $C$ of nilpotency class at most $2$ with $|G:C|\le2^{2(1+\log n)^2(13+10\log n)}$ and $|Z(C):Z(G)|\le2^{4(1+\log n)^3(13+10\log n)}$. Lemma 3.4 and the Sylow decomposition of nilpotent groups reduce the rest to $p$-groups of class $2$, handled in Theorem 5.4 (p. 293) with the general lemmas of Section 4. In Section 7 (p. 294) the paper notes Lemma 7.1, that $G$ is covered by at most $|G:A|\,n(G)$ abelian subgroups for any abelian subgroup $A$, suggests that probably $|G:A|\le2^{4n}$ for a maximal abelian subgroup without proving it, and remarks that it is tempting to conjecture that the index of the centre is largest for extraspecial $2$-groups.

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Results.

• Theorem (p. 287) and Theorem 6.1 (p. 294): $|G:Z(G)|\le c^n$, explicitly $|G:Z(G)|\le2^{2^{25}n}2^{3(2+2\log n)^5}$.
• Corollary (p. 287): $cc(\Gamma)\le c^n$.
• Example (p. 288): extraspecial $S$ of order $2^{2m+1}$ has $n(S)=2m+1$, $|S:Z(S)|=2^{2m}$ and $cc(\Gamma(S))\ge2^m+1$.
• Lemma 3.1 (p. 288): $k\le4n^2$.

Bears on. #117: a set of pairwise commuting elements lies in an abelian subgroup, so the Corollary covers every group with $n(G)\le n$ by at most $c^n$ abelian subgroups, an upper bound $c^n$ for the problem's $h(n)$; the Example gives groups with $n(S)=2m+1$ that need at least $2^m+1$ abelian subgroups. The paper does not determine the base of the exponential.

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