Héthelyi and Külshammer showed that the number of conjugacy classes $k(G)$ of any solvable finite group $G$ whose order is divisible by the square of a prime $p$ is at least $(49p+1)/60$. Here an asymptotic generalization of this result is established. It is proved that there exists a constant $c>0$ such that for any finite group $G$ whose order is divisible by the square of a prime $p$ we have $k(G) \geq cp$.
In this paper, we show that if $p$ is a prime and $G$ is a $p$-solvable group, then $| G:O_p (G) |_p \le (b(G)^p/p)^{1/(p-1)}$ where $b(G)$ is the largest character degree of $G$. If $p$ is an odd prime that is not a Mersenne prime or if the nilpotence class of a Sylow $p$-subgroup of $G$ is at most $p$, then $| G:O_p (G) |_p \le b(G)$.
In this note, we prove: \medskip \noindent {\bf Theorem A:} \emph{ There is a fixed constant $C$ such that for any positive integer $n$ and prime $p$, every finite subgroup $G$ of order coprime to $p$ of ${\rm GL}(n,\mathbb{C})$ has an Abelian normal subgroup $A$ with $$[G:A] \leq (Cp)^{n-1}.$$}
In this paper we study prime graphs of finite groups. The prime graph of a finite group $G$, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing $|G|$} and an edge $p$-$q$ if and only if there exists an element of order $pq$ in $G$. In finite group theory, studying the prime graph of a group has been an important topic for the past almost half century. Only recently prime graphs of solvable groups have been characterized in graph theoretical terms only. In this paper, we continue this line of research and give complete characterizations of several classes of …
Let k(B_0) and l(B_0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B_0 of a finite group G. We prove that, if k(B_0)-l(B_0)=1, then l(B_0)\geq p-1 or else p=11 and l(B_0)=9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B_0)\geq p-1 or else p = 11 and G/O_{p'}(G) =11^2:SL(2,5). These results are useful in the study of principal blocks with a few characters. We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer charact…
Denote by $m(G)$ the largest size of a minimal generating set of a finite group $G$. We estimate $m(G)$ in terms of $\sum_{p\in π(G)}d_p(G),$ where we are denoting by $d_p(G)$ the minimal number of generators of a Sylow $p$-subgroup of $G$ and by $π(G)$ the set of prime numbers dividing the order of $G$.
We give a complete list of all the 70 class two groups of exponent p (p>2) and order p^k for k<9. For each of these groups the number of conjugacy classes is a polynomial in p, and the order of the automorphism group is a polynomial in p. In contrast, Marcus du Sautoy and Michael Vaughan-Lee have given an example of a class two group of exponent p and order p^9 for which neither the number of conjugacy classes, nor the order of the automorphism group is polynomial on residue classes (PORC).
Let $U = \mathbf U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G = \mathbf G(q)$. In [GR}] Röhrle and the second author determined a parameterization of the conjugacy classes of $U$, for $\mathbf G$ of small rank when $q$ is a power of a good prime for $\mathbf G$. As a consequence they verified that the number $k(U)$ of conjugacy classes of $U$ is given by a polynomial in $q$ with integer coefficients. In the present paper, we consider the case when $p$ is a bad prime for $\mathbf G$. We obtain a parameterization of the conjugacy classes of $U$, when $\mathbf G$ has rank less tha…
Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. The degree graph $Δ(G)$ of $G$ is defined as the simple undirected graph whose vertex set ${\rm{V}}(G)$ consists of the prime divisors of the numbers in ${\rm{cd}}(G)$, two distinct vertices $p$ and $q$ being adjacent if and only if $pq$ divides some number in ${\rm{cd}}(G)$. In this note, we provide an upper bound on the size of ${\rm{V}}(G)$ in terms of the clique number $ω(G)$ (i.e., the maximum size of a subset of ${\rm{V}}(G)$ inducing a complete subgraph) of $Δ(G)$. N…
Given a finite root system $Φ$, we show that there is an integer $c=c(Φ)$ such that $\dim\Ext_G^1(L,L')<c$, for any reductive algebraic group $G$ with root system $Φ$ and any irreducible rational $G$-modules $L,L'$. There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for $\Ext^n$, for any integer $n\geq 0$, using a constant depending only on $n$ and the root system. Weaker versions of this are proved in the algebraic and finite group cases, sufficient to give similar r…
In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an optimal way. This is called an {\sl adapted} homotopy basis and there is a corresponding {\sl adapted presentation}. We also give a necessary and sufficient condition for a prime order symplectic matrix to be the image of a prime order element in the mapping-class group.
Tournaments are graphs obtained by assigning a direction for every edge in an undirected complete graph. We give a formula for the number of isomorphism classes of vertex-transitive tournaments with prime order. For that, we introduce Cayley tournaments, which are special Cayley digraphs, and show that there is a one to one identification between the isomorphism classes of vertex-transitive tournaments with prime order and the isomorphism classes of Cayley tournaments with prime order.
In this paper we classify the finite groups satisfying the following property $P_4$: their orders of representatives are set-wise relatively prime for any 4 distinct non-central conjugacy classes.
Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple simply-connected algebraic group over $k$ that is defined and split over the prime field $\mathbb{F}_p$. In this paper we investigate situations where the dimension of a rational cohomology group for $G$ can be bounded by a constant times the dimension of the coefficient module. We then demonstrate how our results can be applied to obtain effective bounds on the first cohomology of the symmetric group. We also show how, for finite Chevalley groups, our methods permit significant improvements over previo…
We show that if $p$ is a prime and $G$ is a finite $p$-solvable group satisfying the condition that a prime $q$ divides the degree of no irreducible $p$-Brauer character of $G$, then the normalizer of some Sylow $q$-subgroup of $G$ meets all the conjugacy classes of $p$-regular elements of $G$.
In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We consider $η$ and direct and semi-direct products. We characterize the normal subgroups $N$ so that $η(G/N) = η(G)$. We set $G^- = \{ g \in G \mid \langle g \rangle {\rm ~is~not ~maximal~cyclic} \}$. We show if $\langle G^- \rangle < G$, then $G/\langle G^- \rangle$ is either (1) an elementary abelian $p$-group for some prime $p$, (2) a Frobenius group whose Frobenius kernel is a $p$-group of exponent $p$ and a Frobenius complement has order $q$ for distinct primes $p$ and $q$, or (3) isomo…
When $G$ is solvable group, we prove that the number of conjugacy classes of elements of prime power order is less than or equal to the number of irreducible characters with values in fields where $\mathbb {Q}$ is extended by prime power roots of unity. We then combine this result with a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order to bound the order of a solvable group by the number of irreducible characters with values in fields extended by prime power roots of unity. This yields for so…
The commuting graph of a group $G$ is the simple undirected graph whose vertices are the non-central elements of $G$ and two distinct vertices are adjacent if and only if they commute. It is conjectured by Jafarzadeh and Iranmanesh that there is a universal upper bound on the diameter of the commuting graphs of finite groups when the commuting graph is connected. In this paper we determine upper bounds on the diameter of the commuting graph for some classes of groups to rule them out as possible counterexamples to this conjecture. We also give an example of an infinite family of groups with tr…
The classical prime geodesic theorem (PGT) gives an asymptotic formula (as $x$ tends to infinity) for the number of closed geodesics with length at most $x$ on a hyperbolic manifold $M$. Closed geodesics correspond to conjugacy classes of $π_1(M)=Γ$ where $Γ$ is a lattice in $G=SO(n,1)$. The theorem can be rephrased in the following format. Let $X(\Z,Γ)$ be the space of representations of $\Z$ into $Γ$ modulo conjugation by $Γ$. $X(\Z,G)$ is defined similarly. Let $π: X(\Z,Γ)\to X(\Z,G)$ be the projection map. The PGT provides a volume form $vol$ on $X(\Z,G)$ such that for sequences of subsets…
