arXiv subjects

Thomas Michael Keller

Publications and source records attributed to Thomas Michael Keller.

At least 19 recordsLinked to original sources

arXiv2026

Large orbits of $p$-groups on conjugacy classes

Let $P$ be a $p$-group that acts faithfully and coprimely on a finite group $H$. A classical result of D.\ S.\ Passman, in the form proved by I.\ M.\ Isaacs, asserts that there exists an element $x\in H$ such that $|C_P(x)|\le (|P|/p)^{1/p}$. In this paper, we study the corresponding question for the action of $P$ on the conjugacy classes of $H$. We show that for every odd prime $p$ the analogue of Isaacs' bound fails for classes. On the other hand, we prove that if $H$ is solvable, then there always exists a conjugacy class $c$ of $H$ whose stabilizer in $P$ has order at most $|P|^{1/2}$, and that this bound is best possible for every prime $p$. As a consequence, there always exists an irreducible character of $H$ whose inertia group in $P$ has order at most $|P|^{1/2}$. The proof proceeds via a result on linear actions of $p$-nilpotent groups, and in the quasiprimitive case it relies on a classification of Holt and Yang and on computer calculations.

math.GR↗

arXiv2026

Linear $2$-Group Actions With Exactly One Regular Orbit

Let $G$ be a finite $2$-group and $V$ a finite faithful irreducible $G$-module. We show that if $G$ has exactly one regular orbit on $V$, then the characteristic $q$ of $V$ is a Fermat prime, $V$ is irreducible over ${\Bbb F}_q$, ${\rm dim}_{{\Bbb F}_q}V$ is a power of $2$, and the centraliser of $G$ in ${\rm GL}(V)$ embeds into $G$. For every Fermat prime $q$ and every $m\ge 1$ we determine the largest orbit size $2^{x_m}$ of a Sylow $2$-subgroup $P_m$ of ${\rm GL}_{2^{m-1}}(q)$ on its natural module $V_m$, and the number of orbits of this size, which is $1$, $2$, or $3$. Our main result is that for every Fermat prime $q$ and every $m\ge 1$ there is an irreducible subgroup of $P_m$ of order $2^{x_m}$ that has exactly one regular orbit on $V_m$. In this respect $p=2$ behaves like $p=3$, and unlike the Mersenne primes $p\ge 7$ studied in an earlier paper. The proof places a group related to a Singer cycle into every block of $q-1$ coordinates. This reduces the problem, for all Fermat primes at once, to a question about subgroups of a Sylow $2$-subgroup of a symmetric group acting on subsets, which we answer by means of fibre products. We also determine which largest orbits occur in two small levels, and we classify the examples in some small cases with the help of a computer.

math.GR↗

arXiv2026

Linear Group Actions with Exactly One Regular Orbit

Let $p$ be an odd prime, $G$ a finite $p$-group, and $V$ a finite faithful irreducible $G$-module. We show that if $G$ has exactly one regular orbit on $V$, then $p=2^{n}-1$ is a Mersenne prime and $(G,V)$ embeds into $(P_m,V_m)$ for some $m\ge 1$, where $P_m$ is a Sylow $p$-subgroup of the monomial group ${\rm GL}_1(2^{n})\wr S_{p^{m-1}}$ and $V_m$ is its natural module, regarded over ${\Bbb F}_2$. We determine the largest orbit size $p^{x_m}$ of $P_m$ on $V_m$ and the number of orbits of this size, and we construct subgroups of $P_m$ of order $p^{x_m}$ whose unique regular orbit is a largest orbit of $P_m$; for $p=3$ such subgroups exist for every $m$, whereas for Mersenne primes $p\ge 7$ we prove that they exist only for $m\ge 3$. Finally, we classify the subgroups of $P_m$ of order $p^{x_m}$ with exactly one regular orbit for $m=2$ and, with the help of a computer, for $p=3$ and $m=3$, where there are $13$ such pairs up to isomorphism; for $p=3$ and $m=4$ we show that there are exactly $4914$ conjugacy classes of such subgroups in $P_4$. In particular, such a subgroup is in general not determined by its regular orbit.

math.GR↗

arXiv2026

On Huppert's $ρ$-$σ$ conjecture for primes in the orders of elements

Let $G$ be a finite group, let $ρ(G)$ denote the number of distinct prime divisors of $|G|$, and let $σ(G)$ be the maximal number of distinct prime divisors of the order of an element of $G$. Let $ρ(n)=\max\{ρ(G) \mid G \text{ solvable}, σ(G)=n\}$. We prove that $ρ(n)\le 4n-1+π(2n)$ for all $n$, and, with the help of a computer, that $ρ(n)\le 9n/2$ for all $n\le 16$. In particular, $ρ(n)\le\lfloor 9n/2\rfloor+2$ and $ρ(n)\le (101/22)n$ for all $n$, and $ρ(n)\le 9n/2$ for all $n\ge 58$. This improves the bound $ρ(n)\le (17/3)n$ due to Yang, and it shows that the bound $ρ(n)\le 5n$ stated by Bellotti, Keller and Trudgian holds, although its original proof contains a gap. Moreover, we show that $ρ(n)\ge 3n$ for all $n\ge 4$, which proves one half of a conjecture of the second author stating that $ρ(n)=3n$ for $n\ge 4$. We also derive a consequence for character codegrees and, conditionally, for arbitrary finite groups. The proofs of the main results do not rely on the classification of finite simple groups.

math.GR↗

arXiv2026

The prime graph complements of $\mathcal{A}_{11}$- and $\mathcal{A}_{13}$-solvable groups

We study prime graph complements of $\mathcal{A}_n$-solvable groups, focusing on the cases $\mathcal{A}_{11}$ and $\mathcal{A}_{13}$. These provide the first classifications of $T$-solvable groups for which $|π(T)|>4$. We develop structural and representation-theoretic tools for analyzing the possible prime graph complements and use them to obtain complete characterizations for both $\mathcal{A}_{11}$-solvable and $\mathcal{A}_{13}$-solvable groups. The methods introduced also provide a framework for studying analogous classification problems for larger alternating groups.

math.GR↗

arXiv2026

On the number of conjugacy classes of finite solvable groups of order divisible by $p^2$

In 2003, Héthelyi and Külshammer proved that if $G$ is a finite solvable group and $p$ is a prime such that $p^2$ divides $|G|$, then $k(G)\geq (49p+1)/60$, and they constructed an infinite family of groups for which this bound was claimed to be attained. We point out that the groups in this family are not Frobenius groups, so that their class numbers are considerably larger than stated, and that in fact the bound is never attained. We then determine the best possible bound of this shape: we show that $k(G)\geq (49p+7)/60$ for all primes $p$, which is sharp for $p=17$, and that $k(G)\geq (49p+451)/60$ for all primes $p>2719$, which is sharp for infinitely many primes $p$. Thus the slope $49/60$ found by Héthelyi and Külshammer is best possible, but the correct additive constant is $451/60$ rather than $1/60$.

math.GR↗

arXiv2026

Abelian quotients and the largest orbit sizes of linear groups of odd order

Let $G$ be a finite nonabelian group of odd order, let $p$ be the smallest prime divisor of $|G|$, and let $V$ be a finite faithful completely reducible $G$-module, possibly of mixed characteristic. Suppose that $M$ is the largest orbit size in the action of $G$ on $V$. It is known that $|G:G'|\le M$, and that equality holds only for abelian groups if $|G|$ is odd. We prove that $|G:G'|\le M/p$, unless $p=3$ and $G$ is the direct product of an abelian group and a group isomorphic to $Γ(2^3)$ or to $Γ(2^3)\times C_3$ acting on $V$ in a specific way, in which case $|G:G'|=3M/7$. In particular, $|G:G'|\le 3M/7$ for every nonabelian group of odd order. If $|V|$ is odd, then $|G:G'|=M/p$ holds if and only if $|G'|=p$; more precisely, nilpotent groups then have regular orbits, whereas non-nilpotent groups satisfy $|G:G'|\le 3M/13$, and even $|G:G'|<M/(2p)$ if $p\ge5$. The bound $3M/13$ is best possible.

math.GR↗

arXiv2026

5-cycles in the complement of minimal prime graphs

Minimal prime graphs (MPGs) are a special class of prime graphs (also known as Gruenberg-Kegel graphs) associated with finite solvable groups. A graph is an MPG if it has at least two vertices, is connected, its complement is triangle-free and 3-colorable, and the addition of an edge to the complement will violate triangle-freeness or 3-colorability. In this paper, we continue the study of the complements of MPGs focusing on their cycle structure. Our main result establishes that every edge in the complement of an MPG is contained in a 5-cycle. This finding is a much stronger form of an older result stating that every minimal prime graph complement contains at least one induced 5-cycle.

math.CO↗

arXiv2025

Degree 2 vertices in minimal prime graph complements

Minimal prime graphs are connected graphs on at least two vertices whose complements satisfy the following conditions: triangle-freeness, 3-colorability, and edge-maximality with respect to the latter two properties. These graphs are prime graphs (or Gruenberg-Kegel graphs) of finite solvable groups with the maximum number of Frobenius actions among their Sylow subgroups, and as such minimal prime graph complements have been shown to be highly structured, including, for instance, the presence of induced 5-cycles. It is also known that the minimum degree of minimal prime graph complements is 2. In this note, we show that the existence of a degree 2 vertex in a minimal prime graph complement determines its whole structure: it is simply a 5-cycle with three vertices, exactly two of which are adjacent to each other, being duplicated finitely often. In particular, such graphs belong to a class of graphs known as reseminant.

math.CO↗

arXiv2025

Realizing Gruenberg-Kegel graphs of $T$-solvable groups with structurally simplified extensions of $T$

Given a finite group $G$, its prime graph $Γ(G)$ (also known as its Gruenberg-Kegel graph) is the graph whose vertices are the prime divisors of $|G|$ and where edges $\{p, q\}$ exist whenever $G$ contains an element of order $pq$. We continue the study of prime graphs for $T$-solvable groups; that is, groups whose composition factors are either abelian or isomorphic to some fixed non-abelian simple group $T$. For a large class of non-abelian simple groups $T$, we prove that the prime graph complements of $T$-solvable groups are always realizable by a solvable group and a quasi simple or almost simple $T$-solvable group acting by automorphisms on a direct product of elementary abelian groups. We conjecture that a similar result holds in full generality. Moreover, we apply our result to classify in purely graph-theoretic terms the prime graph complements of $\operatorname{PSL}(2,13)$-solvable groups, and indicate other interesting classes of groups matching the assumptions of our main theorem.

math.GR↗

arXiv2025

Criteria for Classifying Prime Graphs of PSL(2, q)-Solvable Groups

For a finite group $G$, the prime graph $Γ(G)$ (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide $|G|$ such that two vertices $p$ and $q$ share an edge if and only if there is an element of order $pq$ in $G$. The prime graphs of solvable groups have been classified. The prime graphs of groups whose noncyclic composition factors are isomorphic to a single nonabelian simple group $T$ where $|T|$ is divisible by three or four distinct primes have been classified except for the cases where $T = \operatorname{PSL}(2,q)$ for $q\neq 2^5$ and $|\operatorname{PSL}(2,q)|$ is divisible by exactly four primes. In this paper, we provide criteria for general classification results for certain classes of $T$, and then use them to classify the prime graphs of some $T$-solvable groups for $T$ a suitably small $\operatorname{PSL}(2, q)$-group. We also provide general results on the prime graphs of $T$-solvable groups where $T$ is a member of the possibly infinite family of groups $\operatorname{PSL}(2, 2^f)$ such that $f\geq 5, f$ is prime, and $|\operatorname{PSL}(2, 2^f)|$ is divisible by exactly four primes. This is the first paper to prove general results about the prime graphs of $T$-solvable groups where $T$ belongs to a large (probably infinite) family of groups.

math.GR↗

arXiv2025

More on Landau's theorem and Conjugacy Classes

In this paper we present two new results on the number of certain conjugacy classes of a finite group. For a finite group $G$, let $n(G)$ be the maximum of $k_{p}(G)$ taken over all primes $p$ where $k_{p}(G)$ denotes the number of conjugacy classes of nontrivial $p$-elements in $G$. Using a recent theorem of Giudici, Morgan and Praeger, we prove that there exists a function $f(x)$ with $f(x) \to \infty$ as $x \to \infty$ such that $n(G) \geq f(|G|)$ for any finite group $G$. Let $G$ be a finite group, and let $p$ be a prime dividing $|G|$. Let $k_{p'}(G)$ denote the number of conjugacy classes of elements of $G$ whose orders are coprime to $p$. We show that either $p=11$ and $G=C_{11}^2\rtimes \text{\rm SL}(2,5)$, or there exists a factorization $p-1 = ab$ with $a$ and $b$ positive integers, such that $k_{p}(G) \geq a$ and $k_{p'}(G) \geq b$ with equalities in both cases if and only if $G=C_p \rtimes C_b$ with $C_G(C_p) = C_p$.

math.GR↗

arXiv2024

Classification of the Prime Graphs of $\operatorname{Sz}(8)$-, $\operatorname{Sz}(32)$-, and $\operatorname{PSL}(2, 2^5)$-Solvable Groups

For a finite group $G$, the vertices of the prime graph $Γ(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if there is an element of order $pq$ in $G$. Prime graphs of solvable groups have been classified, and prime graphs of groups whose noncyclic composition factors are isomorphic to a single nonabelian simple group $T$ have been classified in the case where $T$ has order divisible by exactly three or four distinct primes, except for the cases $T = \operatorname{Sz}(8)$, $T = \operatorname{Sz}(32)$, and $T = \operatorname{PSL}(2,q)$, which in some sense are the hardest cases. In this paper, we complete the classification for $T = \operatorname{Sz}(32)$, $T = \operatorname{Sz}(8)$, and $T = \operatorname{PSL}(2,2^5)$, with the latter two being the first cases ever studied where $|\text{Out}(T)|$ has prime factors which do not divide $|T|$. The groups studied in this paper are also the first ones requiring knowledge of their Brauer character tables to complete the classification task.

math.GR↗

arXiv2024

Classifying prime graphs of finite groups -- a methodical approach

For a finite group $G$, the vertices of the prime graph $Γ(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in $G$. Prime graphs of solvable groups as well as groups whose noncyclic composition factors have order divisible by exactly three distinct primes have been classified in graph-theoretic terms. In this paper, we begin to develop a general theory on the existence of edges in the prime graph of an arbitrary $T$-solvable group, that is, a group whose composition factors are cyclic or isomorphic to a fixed nonabelian simple group $T$. We then apply these results to classify the prime graphs of $T$-solvable groups for, in a suitable sense, most $T$ such that $|T|$ has exactly four prime divisors. We find that these groups almost always have a 3-colorable prime graph complement containing few possible triangles.

math.GR↗

arXiv2023

A new lower bound for the number of conjugacy classes

In 2003, Héthelyi and Külshammer proposed that if $G$ is a finite group and $p$ is a prime dividing the group order, then $k(G)\geq 2\sqrt{p-1}$, and they proved this conjecture for solvable $G$ and showed that it is sharp for those primes $p$ for which $\sqrt{p-1}$ is an integer. This initiated a flurry of activity, leading to many generalizations and variations of the result; in particular, today the conjecture is known to be true for all finite groups. In this note, we put forward a natural new and stronger conjecture, which is sharp for all primes $p$, and we prove it for solvable groups, and when $p$ is large, also for arbitrary groups.

math.GR↗

arXiv2023

Prime divisors and the number of conjugacy classes of finite groups

We prove that there exists a universal constant $D$ such that if $p$ is a prime divisor of the index of the Fitting subgroup of a finite group $G$, then the number of conjugacy classes of G is at least $Dp/log_2 p$. We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1/3$.

math.GR↗

arXiv2022

The Prime Graphs of Groups With Arithmetically Small Composition Factors

We continue the study of prime graphs of finite groups, also known as Gruenberg-Kegel graphs. The vertices of the prime graph of a finite group are the prime divisors of the group order, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in the group. Prime graphs of solvable groups have been characterized in graph theoretical terms only, as have been the prime graphs of groups whose only nonsolvable composition factor is $A_5$. In this paper we classify the prime graphs of all groups whose composition factors have arithmetically small orders, that is, have no more than three prime divisors in their orders. We find that all such graphs have $3$-colorable complements, and we provide full characterizations of the prime graphs of such groups based on the exact type and multiplicity of the nonabelian composition factors of the group.

math.GR↗

arXiv2022

A classification of prime graphs of pseudo-solvable groups

The prime graph $Γ(G)$ of a finite group $G$ (also known as the Gruenberg-Kegel graph) has as its vertices the prime divisors of $|G|$, and $p\text-q$ is an edge in $Γ(G)$ if and only if $G$ has an element of order $pq$. Since their inception in the 1970s these graphs have been studied extensively; however, completely classifying the possible prime graphs for larger families of groups remains a difficult problem. For solvable groups such a classification was found in 2015. In this paper we go beyond solvable groups for the first time and characterize prime graphs of a more general class of groups we call pseudo-solvable. These are groups whose composition factors are either cyclic or $A_5$. The classification is based on two conditions: the vertices $\{2,3,5\}$ form a triangle in $\overlineΓ(G)$ or $\{p,3,5\}$ form a triangle for some prime $p\neq 2$.

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