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Let $G$ be a finite group, and let $\text{Irr}(G)$ denote the set of the irreducible complex characters of $G$. An element $g\in G$ is called a vanishing element of $G$ if there exists $\chi\in\text{Irr}(G)$ such that $\chi(g)=0$ (i.e., $g$ is a zero of $\chi$) and, in this case, the conjugacy class $g^G$ of $g$ in $G$ is called a vanishing conjugacy class. In this paper we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group $G$ such that every non-linear $\chi\in\text{Irr}(G)$ vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero.

For a finite group $G$ and an irreducible complex character $\chi$ of $G$, the codegree of $\chi$ is defined by $\textrm{cod}(\chi)=|G:\textrm{ker}(\chi)|/\chi(1)$, where $\textrm{ker}(\chi)$ is the kernel of $\chi$. In this paper, we show that if $H$ is a finite simple exceptional group of Lie type or a projective special linear group and $G$ is any finite group such that the character codegree sets of $G$ and $H$ coincide, then $G$ and $H$ are isomorphic., Comment: 18 pages

Let $G$ be a finite group and let $\textrm{cd}(G)$ be the set of all complex irreducible character degrees of $G.$ In this paper, we show that if $\textrm{cd}(G)=\textrm{cd}(H),$ where $H$ is a finite simple exceptional group of Lie type, then $G\cong H\times A,$ where $A$ is an abelian group. This completes the verification of Huppert's Conjecture for all finite simple exceptional groups of Lie type., Comment: 18 pages. Some typos are corrected

Let $G$ be a finite group and let $H$ be a subgroup of $G$. We say that $H$ is extremely closed in $G$ if $\langle H,H^g\rangle\cap N_G(H)=H$ for all $g\in G.$ In this paper, we determine the structure of finite groups with an extremely closed abelian $p$-subgroup for some prime $p$. In particular, we show that if $G$ contains such a subgroup $H$, then $G=N_G(H)O_{p'}(G).$ This is a variant on the celebrated Glauberman's $Z^*$-Theorem., Comment: 19 pages; to appear in Pacific J. Math

Thompson's theorem stated that a finite group $G$ is solvable if and only if every $2$-generated subgroup of $G$ is solvable. In this paper, we prove some new criteria for both solvability and nilpotency of a finite group using certain condition on $2$-generated subgroups. We show that a finite group $G$ is solvable if and only if for every pair of two elements $x$ and $y$ in $G$ of coprime prime power order, if $\langle x,y\rangle$ is solvable, then $\langle x,y^g\rangle$ is solvable for all $g\in G$. Similarly, a finite group $G$ is nilpotent if and only if for every pair of elements $x$ and $y$ in $G$ of coprime prime power order, if $\langle x,y\rangle$ is solvable, then $x$ and $y^g$ commute for some $g\in G.$ Some applications to graphs defined on groups are given., Comment: 23 pages; comments welcome

A subgroup $R$ of a finite group $G$ is weakly subnormal in $G$ if $R$ is not subnormal in $G$ but it is subnormal in every proper overgroup of $R$ in $G$. In this paper, we first classify all finite groups $G$ which contains a weakly subnormal $p$-subgroup for some prime $p$. We then determine all finite groups containing a cyclic weakly subnormal $p$-subgroup. As applications, we prove a number of variations of the Baer-Suzuki theorem using the orders of certain group elements., Comment: 25 pages; comments welcome

In this paper, we determine the finite groups with a Sylow $r$-subgroup contained in a unique maximal subgroup. The proof involves a reduction to almost simple groups, and our main theorem extends earlier work of Aschbacher in the special case $r=2$. Several applications are presented. This includes some new results on weakly subnormal subgroups of finite groups, which can be used to study variations of the Baer-Suzuki theorem., Comment: 50 pages; to appear in Advances in Mathematics

We classify all finite groups $G$ which possesses an element $x\in G$ such that every irreducible character of $G$ takes a root of unity value at $x$.

In this paper, we study the proportion of vanishing elements of finite groups. We show that the proportion of vanishing elements of every finite non-abelian group is bounded below by $1/2$ and classify all finite groups whose proportions of vanishing elements attain this bound. For symmetric groups of degree at least $5$, we show that this bound is at least $2327/2520$ which is best possible.

In this paper, we study the structure of finite groups with a large number of conjugacy classes of $p$-elements for some prime $p$. As consequences, we obtain some new criteria for the existence of normal $p$-complements in finite groups.

We prove that there exists an integer-valued function f on positive integers such that if a finite group G has at most k real-valued irreducible characters, then |G/Sol(G)| is at most f(k), where Sol(G) denotes the largest solvable normal subgroup of G. In the case k = 5, we further classify G/Sol(G). This partly answers a question of Iwasaki [15] on the relationship between the structure of a finite group and its number of real-valued irreducible characters., Comment: 13 pages

We investigate the structure of finite groups whose non-central real class sizes have the same $2$-part. In particular, we prove that such groups are solvable and have $2$-length one. As a consequence, we show that a finite group is solvable if it has two real class sizes. This confirms a conjecture due to G. Navarro, L. Sanus and P. Tiep., Comment: 15 pages

In this paper, we study the structures of finite groups using some arithmetic conditions on the sizes of real conjugacy classes. We prove that a finite group is solvable if the prime graph on the real class sizes of the group is disconnected. Moreover, we show that if the sizes of all non-central real conjugacy classes of a finite group $G$ have the same $2$-part and the Sylow $2$-subgroup of $G$ satisfies certain condition, then $G$ is solvable., Comment: 13 pages

In this paper, we study some variations of the well-known It\^{o}-Michler theorem for $p$-Brauer characters using various inequalities involving the $p$-Brauer character degrees of finite groups. Several new criteria for the existence of a normal Sylow $p$-subgroup of finite groups are obtained using the $p$-parts and $p'$-parts of the $p$-Brauer character degrees., Comment: Statements of Claim (1) in Theorems 2.6 and 3.3 are corrected

Chillag has showed that there is a single generalization showing that the sums of ordinary character tables, Brauer character, and projective indecomposable characters are positive integers. We show that Chillag's construction also applies to Isaacs' $\pi$-partial characters. We show that if an extra condition is assumed, then we can obtain upper and lower bounds on the Chillag's table sums. We will demonstrate that this condition holds for Brauer characters, $\pi$-partial characters, and projective indecomposable characters, and so we obtain upper and lower bounds for the table sums in those cases. We also obtain results regarding the sums of rows and columns in these tables., Comment: 10 pages

Let $G$ be a finite group, and write ${\rm cd}(G)$ for the degree set of the complex irreducible characters of $G$. The group $G$ is said to satisfy the {\it two-prime hypothesis} if, for any distinct degrees $a, b \in {\rm cd}(G)$, the total number of (not necessarily different) primes of the greatest common divisor ${\rm gcd}(a, b)$ is at most $2$. In this paper, we prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to ${\rm PSL}_2 (q)$ for $q \geq 7$., Comment: 18 pages

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 $G$., Comment: A shorter version of this paper will appear in Proceedings of the AMS

We prove that for every $\epsilon > 0$ there exists a $\delta > 0$ so that every group of order $n \geq 3$ has at least $\delta \log_{2} n/{(\log_{2} \log_{2} n)}^{3+\epsilon}$ conjugacy classes. This sharpens earlier results of Pyber and Keller. Bertram speculates whether it is true that every finite group of order $n$ has more than $\log_{3}n$ conjugacy classes. We answer Bertram's question in the affirmative for groups with a trivial solvable radical.

We prove that the alternating groups of degree at least $5$ are uniquely determined up to an abelian direct factor by the degrees of their irreducible complex representations. This confirms Huppert's Conjecture for alternating groups., Comment: 20 pages

Let $G$ be a transitive permutation group on a finite set of size at least $2$. By a well known theorem of Fein, Kantor and Schacher, $G$ contains a derangement of prime power order. In this paper, we study the finite primitive permutation groups with the extremal property that the order of every derangement is an $r$-power, for some fixed prime $r$. First we show that these groups are either almost simple or affine, and we determine all the almost simple groups with this property. We also prove that an affine group $G$ has this property if and only if every two-point stabilizer is an $r$-group. Here the structure of $G$ has been extensively studied in work of Guralnick and Wiegand on the multiplicative structure of Galois field extensions, and in later work of Fleischmann, Lempken and Tiep on $r'$-semiregular pairs., Comment: 30 pages; to appear in Manuscripta Math

Let $G$ be a finite primitive permutation group and let $\kappa(G)$ be the number of conjugacy classes of derangements in $G$. By a classical theorem of Jordan, $\kappa(G) \geqslant 1$. In this paper we classify the groups $G$ with $\kappa(G)=1$, and we use this to obtain new results on the structure of finite groups with an irreducible complex character that vanishes on a unique conjugacy class. We also obtain detailed structural information on the groups with $\kappa(G)=2$, including a complete classification for almost simple groups., Comment: 29 pages

We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let $n\geq 5$ be an integer, $G$ a finite group, and let $\AAA$ and $\SSS^\pm$ denote the double covers of $\Al_n$ and $\Sy_n$, respectively. We prove that $\CC G\cong \CC \AAA$ if and only if $G\cong \AAA$, and $\CC G\cong \CC \SSS^+\cong\CC\SSS^-$ if and only if $G\cong \SSS^+$ or $\SSS^-$. This in particular completes the proof of a conjecture proposed by the second and fourth authors that every finite quasi-simple group is determined uniquely up to isomorphism by the structure of its complex group algebra. The known results on prime power degrees and relatively small degrees of irreducible (linear and projective) representations of the symmetric and alternating groups together with the classification of finite simple groups play an essential role in the proofs., Comment: 27 pages, the previous version is revised slightly, to appear in Algebra & Number Theory

Let G be a finite group and let Irr(G) be the set of all irreducible complex characters of G. Let cd(G) be the set of all character degrees of G and denote by \rho(G) the set of primes which divide some character degrees of G. The prime graph \Delta(G) associated to G is a graph whose vertex set is \rho(G) and there is an edge between two distinct primes p and q if and only if the product pq divides some character degree of G. In this paper, we show that the prime graph \Delta(G) of a finite group G is 3-regular if and only if it is a complete graph with four vertices., Comment: 18 pages

Let G be a finite group. An element x in G is a real element if x is conjugate to its inverse in G. For x in G, the conjugacy class x^G is said to be a real conjugacy class if every element of x^G is real. We show that if 4 divides no real conjugacy class size of a finite group G, then G is solvable. We also study the structure of such groups in detail. This generalizes several results in the literature., Comment: 9 pages

Let G be a finite group and let cd(G) be the set of all complex irreducible character degrees of G Let \rho(G) be the set of all primes which divide some character degree of G. The prime graph \Delta(G) attached to G is a graph whose vertex set is \rho(G) and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of G. In this paper, we show that if G is a finite group whose prime graph \Delta(G) has no triangles, then \Delta(G) has at most 5 vertices. We also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occur as prime graphs of some finite groups. Finally, we show that the prime graph of a finite group can never be a cycle nor a tree with at least 5 vertices., Comment: 13 pages

In this paper, we will show that if for every nonlinear complex irreducible character of a finite group G, some multiple of it is induced from an irreducible character of some proper subgroup of G, then G is solvable. This is a generalization of Taketa's Theorem on the solvability of M-group., Comment: 17 pages, to appear in J. Algebra

Let $G$ be a finite group and let $cd(G)$ be the set of all irreducible complex character degrees of $G$. It was conjectured by Huppert in Illinois J. Math. 44 (2000) that, for every non-abelian finite simple group $H$, if $cd(G)=cd(H)$ then $G\cong H\times A$ for some abelian group $A$. In this paper, we confirm the conjecture for the family of projective special linear groups $\textrm{PSL}_4(q)$ with $q\geq 13$., Comment: 26 pages

In this paper, we will show that nonabelian simple classical groups of Lie type are uniquely determined by the structure of their complex group algebras., Comment: 10 pages, to appear in Journal of Algebra

Let $G$ be a finite group. Denote by $\textrm{Irr}(G)$ the set of all irreducible complex characters of $G.$ Let $\textrm{cd}(G)=\{\chi(1)\;|\;\chi\in \textrm{Irr}(G)\}$ be the set of all irreducible complex character degrees of $G$ forgetting multiplicities, and let $\textrm{X}_1(G)$ be the set of all irreducible complex character degrees of $G$ counting multiplicities. Let $H$ be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if $S$ is a non-abelian simple group and $\textrm{cd}(S)\subseteq \textrm{cd}(H)$ then $S$ must be isomorphic to $H.$ As a consequence, we show that if $G$ is a finite group with $\textrm{X}_1(G)\subseteq \textrm{X}_1(H)$ then $G$ is isomorphic to $H.$ In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras., Comment: 18 pages

Let $H\leq K$ be subgroups of a group G. We say that H is strongly closed in K with respect to G if whenever $a^g \in K$ where $a \in H, g \in G,$ then $a^g \in H.$ In this paper, we investigate the structure of a group G under the assumption that every subgroup of order $2^m$ (and 4 if m = 1) of a 2- Sylow subgroup S of G is strongly closed in S with respect to G. Some results related to 2-nilpotence and supersolvability of a group G are obtained. This is a complement to Guo and Wei (J. Group Theory 13 (2010), no. 2, 267-276)., Comment: 6 pages

Let N be a minimal normal nonabelian subgroup of a finite group G. We will show that there exists a nontrivial irreducible character of N of degree at least 5 which is extendible to G. This result will be used to settle two open questions raised by Berkovich and Mann, and Berkovich and Zhmud'., Comment: 5 pages

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever K is normal in M, then K^G\cap M=K, where K^G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every maximal subgroup of G is an NR -subgroup then G is solvable. This gives a positive answer to a conjecture posed in Berkovich (Houston J Math 24:631-638, 1998)., Comment: 5 pages

In this paper we classify all maximal subgroups M of a nearly simple primitive rank 3 group G of type L=Omega_{2m+1}(3), m > 3; acting on an L-orbit E of non-singular points of the natural module for L such that 1_P^G <=1_M^G where P is a stabilizer of a point in E. This result has an application to the study of minimal genera of algebraic curves which admit group actions., Comment: 41 pages, to appear in Forum Mathematicum

In this paper, we will prove some sufficient conditions for the solvability of groups., Comment: 9 pages, to appear in Communications in Algebra