Class numbers and nilpotent subgroups of $\mathrm{PGL}(2,q)$

Abstract: We show that for certain odd prime powers $q$, the number of conjugacy classes of $\mathrm{PGL}(2,q)$ is greater than the order of its largest nilpotent subgroup. This answers negatively a question of Liebeck and Pyber.

• The paper establishes that for PGL(2,q), the number of conjugacy classes can exceed the order of the largest nilpotent subgroup, resolving an open conjecture.
• Using detailed subgroup classifications and Macdonald's enumeration, the authors identify three cases based on prime power q with distinct structural outcomes.
• The work leverages both theoretical analysis and GAP computations to construct infinite families of finite groups where k(G) > n(G), influencing future finite group studies.

Class Numbers and Nilpotent Subgroups of $\mathrm{PGL}(2,q)$

Introduction

This paper investigates the relationship between the number of conjugacy classes, $k(G)$, and the size of the largest nilpotent subgroup, $n(G)$, within the projective general linear groups $G = \mathrm{PGL}(2,q)$, for prime powers $q$. The work provides a comprehensive classification of these quantities, settling a question raised by Liebeck and Pyber regarding the existence of a universal constant $c$ such that $k(G) \leq n(G)^c$ for all finite groups $G$, and in particular whether $c=1$ suffices.

Classification of Nilpotent Subgroups in $\mathrm{PGL}(2,q)$

Using the known subgroup classifications of $k(G)$0 and the relation between $k(G)$1 and $k(G)$2, the paper determines $k(G)$3 as follows:

• For general $k(G)$4, the maximal nilpotent subgroups are abelian, specifically $k(G)$5 unless $k(G)$6 is of the special form $k(G)$7.
• For $k(G)$8, the maximal nilpotent subgroup is dihedral of order $k(G)$9.
• Other subgroup families (including certain dihedral, alternating, and semi-direct product subgroups) are shown to not yield larger nilpotent subgroups due to non-commutativity or lack of appropriate Sylow subgroup structure.

Consequently, the classification of $n(G)$0 is succinct, with nilpotent subgroups tightly constrained by $n(G)$1.

Enumeration of Conjugacy Classes

Building upon Macdonald's explicit enumeration, the paper asserts $n(G)$2. This leads to three cases:

• If $n(G)$3, then $n(G)$4.
• For $n(G)$5, $n(G)$6.
• For $n(G)$7 ($n(G)$8), $n(G)$9.

The last case is of particular interest, as it yields infinite families of groups with more conjugacy classes than the order of their largest nilpotent subgroup, directly answering the open question in the negative.

Implications for Universal Bounds

The main consequence is the disproval of the $G = \mathrm{PGL}(2,q)$0 case in the bound $G = \mathrm{PGL}(2,q)$1 for all finite groups. The paper shows that for certain $G = \mathrm{PGL}(2,q)$2,

$G = \mathrm{PGL}(2,q)$3

which rules out the conjecture $G = \mathrm{PGL}(2,q)$4 universally. This establishes the existence of infinite families of finite groups, starting with $G = \mathrm{PGL}(2,q)$5 and $G = \mathrm{PGL}(2,q)$6, for which the number of conjugacy classes exceeds the maximal nilpotent subgroup order.

Construction of Further Examples

Through direct and subdirect product constructions, especially leveraging the normal subgroup structure $G = \mathrm{PGL}(2,q)$7 for $G = \mathrm{PGL}(2,q)$8 odd, new groups with $G = \mathrm{PGL}(2,q)$9 are systematically generated. Properties of both $q$0 and $q$1 under these product operations are carefully justified, allowing for the explicit construction of larger groups with the desired inequality. Computational verification using GAP confirms the absence of smaller finite group counterexamples, reinforcing the completeness of the classification in the paper.

Further Remarks

Contrast is drawn with $q$2, where class numbers and maximal nilpotent subgroup orders match except for the special cases $q$3, and $q$4 with $q$5 even, where equality always holds.

Conclusion

This work completes the analysis of the class number versus largest nilpotent subgroup problem for $q$6, demonstrating that the natural expectation $q$7 fails outside certain explicit families. The results not only resolve a long-standing question but also clarify the subgroup structure and class arithmetic of $q$8, with implications for the construction and analysis of finite groups possessing large numbers of conjugacy classes relative to nilpotent subgroups. These insights may inform further advances in the algebraic and computational classification of finite groups, and the precise quantification of their internal symmetries.