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Let G be a finite group and N be a nontrivial normal subgroup of G. Let Irr (G | N) be the set of all irreducible characters whose kernel does not contain N as a subgroup and Irr 2 (G | N) be the set of all linear and even degree characters of Irr (G | N) . By acd 2 (G | N) we mean the average degree of irreducible characters of Irr 2 (G | N) . Let Irr 2 (G | N) ≠ ∅ . We prove that N is solvable if acd 2 (G | N) ≤ 5 / 2 . Also, we prove the solvability of G by assuming acd 2 (G | N) < 5 / 2 . Moreover, the bounds are sharp. [ABSTRACT FROM AUTHOR]

We determine the rationality properties of unipotent characters of finite reductive groups arising as fixed points of disconnected reductive groups under a Frobenius map. In the proof, we use realisations of characters in ℓ -adic cohomology groups of Deligne–Lusztig varieties as well as block theoretic considerations. [ABSTRACT FROM AUTHOR]

Let C n and Q n denote the cyclic group and the generalized quaternion group of order n, respectively. We determine all possible values of the integer group determinants of C 8 ⋊ 3 C 2 and Q 8 ⋊ C 2 , which are the unresolved groups of order 16 (Serrano, Paudel and Pinner also obtained a complete description of the integer group determinants of Q 8 ⋊ C 2 independently of this paper and presented it a few days earlier than this paper). Also, we give a diagram of the set inclusion relations between the integer group determinants for all groups of order 16 [ABSTRACT FROM AUTHOR]

AbstractIn this paper, by using the theory of linear limits of complex characters due to Dade and Loukaki, we present a new and relatively simple proof of Lewis’ theorem regarding the minimal inducing degree of an irreducible character of a maximal subgroup of an M-group. [ABSTRACT FROM AUTHOR]

AbstractLet G be a finite group, and g∈G. Then g is said to be a vanishing element of G, if there exists an irreducible character χ of G such that χ(g)=0. Denote by Vo(G) the set of the orders of vanishing elements of G. We say a non-abelian group G is V-recognizable, if any group N with Vo(N)=Vo(G) is isomorphic to G. In this paper, we investigate the V-recognizability of E8(p), where p is a prime number. As an application, among the 610 primes p with p<10000 and p≡0,1,4 ( mod 5), we obtain that the method is always valid for confirming the V-recognizability of E8(p) for all such p but 919, 1289, 1931, 3911, 4691, 5381 and 7589. [ABSTRACT FROM AUTHOR]

Let Kern (G) denote the set of nonlinear irreducible character kernels of a finite group G. In this paper, we classify decomposable groups G with | Kern (G) | ≤ 3 . In particular, nilpotent groups G with | Kern (G) | ≤ 3 are determined. [ABSTRACT FROM AUTHOR]

For an irreducible character χ of a finite group G, the codegree of χ is defined as | G : ker (χ) | / χ (1) . In this paper, we characterize finite p-groups with three character codegrees. [ABSTRACT FROM AUTHOR]

Let χ be an irreducible character of a group G, and S c (G) = ∑ χ ∈ Irr (G) cod (χ) be the sum of the codegrees of the irreducible characters of G. Write fcod (G) = S c (G) | G |. We aim to explore the structure of finite groups in terms of fcod (G). On the other hand, we determine the lower bound of S c (G) for nonsolvable groups and prove that if G is nonsolvable, then S c (G) ⩾ S c (A 5) = 68 , with equality if and only if G ≅ A 5. Additionally, we show that there is a solvable group so that it has the codegree sum as A 5. [ABSTRACT FROM AUTHOR]

Let G be a finite non-abelian simple group, C a non-identity conjugacy class of G, and ΓC the Cayley graph of G based on C ∪ C − 1 . Our main result shows that in any such graph, there is an involution at bounded distance from the identity. [ABSTRACT FROM AUTHOR]

Let G be a finite group and Irr (G) be the set of irreducible characters of G. The codegree of an irreducible character χ of the group G is defined as cod (χ) = | G : ker (χ) | / χ (1) . Let b c (G) be the largest codegree of G. In this paper, we study how the structure of a group G is bounded by its largest codegree b c (G) . Firstly, we give a criterion for solvability: If b c (G) < 20 , then G is solvable. Secondly, we consider the nonsolvable groups G with a small b c (G) and prove that, if b c (G) < 56 , then G is isomorphic to A 5 , S 5 or A 5 × A where A is an elementary abelian 2-group. [ABSTRACT FROM AUTHOR]

AbstractLet o be a compact discrete valuation ring with maximal ideal ℘ such that the finite residue field o/℘ has characteristic p. For r≥2 and p=2, we obtain the branching rules for the restriction of a regular representation of GL2(o/℘r) to SL2(o/℘r). These results have different behavior than that of the known case of p≠2. [ABSTRACT FROM AUTHOR]

AbstractWe give a generic description of the non-defining primes l, for which the Steinberg character lies in the principal l-block of a finite reductive group. [ABSTRACT FROM AUTHOR]

AbstractWe define, and denote by anz(G), the average number of zeros of characters of a finite group G as the number of zeros in the character table of G dividing the number of irreducible characters of G. In this paper, we classify odd-order nonabelian groups G with anz(G)≤2. [ABSTRACT FROM AUTHOR]

AbstractLet C be field of complex numbers and X be a connected quandle of prime order. We study the regular representation of X by describing the quandle ring C[X] as direct sum of right simple ideals. We provide description of tensor product of connected quandles of prime order. We further discuss multiplicity freeness of quandle ring decomposition for connected quandles of order ≤47 and prove that C[X] decomposes multiplicity free for affine connected quandle X. [ABSTRACT FROM AUTHOR]

AbstractFor a character χ of a finite group G, the number cod(χ)=[G:Kerχ]χ(1) is called the codegree of χ. In this paper, we show that if for every non-principal irreducible characters χ and ϕ of G with cod(χ)≠cod(ϕ), the greatest common divisor of cod(χ) and cod(ϕ) is a prime-power, then either G is solvable or G/Sol(G) is isomorphic to one of the groups Alt5 or PSL2(8), where Sol(G) is the solvable radical of the group G. [ABSTRACT FROM AUTHOR]

We consider the central Schur ring associated with the standard supercharacters of the adjoint group G (A) of a finite radical ring A , and define supercharacters of the subgroup C G (A) (σ) consisting of elements fixed by an involution of G (A) naturally defined by an (anti-)involution of A . In particular, we extend known results for unipotent subgroups of the classical finite Chevalley groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A vanishing element of G is an element g ∈ G such that χ (g) = 0 for some irreducible complex character χ of G. We will denote by Van(G) the set of vanishing elements of G and by Vo (G) the set of the order of elements in Van (G) . For certain specified primes p that satisfy p ≡ ± 5 (mod 24) , if Vo (G) = Vo (PSL (2 , p)) , then G ≅ PSL (2 , p) . [ABSTRACT FROM AUTHOR]

For a character χ of a finite group G, the number cod(χ)≔ ∣G: ker(χ)∣/χ(1) is called the codegree of χ. In this paper, we give a solvability criterion for a finite group G depending on the minimum of the ratio χ(1)2/cod(χ), when χ varies among the irreducible characters of G. [ABSTRACT FROM AUTHOR]

For a prime p, we show that uniqueness of factorization into irreducible Σ p 2 -invariant representations of Z / p ≀ Z / p holds if and only if p = 2 . We also show nonuniqueness of factorization for Σ 8 -invariant representations of D 8 ≀ Z / 2 . The representation ring of Σ p 2 -invariant representations of Z / p ≀ Z / p is determined completely when p equals two or three. [ABSTRACT FROM AUTHOR]

Let G be a finite group, F be one of the fields Q , R or C , and N be a non-trivial normal subgroup of G. Let acd F ∗ (G) and acd F , even (G | N) be the average degree of all non-linear F -valued irreducible characters of G and of even degree F -valued irreducible characters of G whose kernels do not contain N, respectively. We assume the average of an empty set is zero for more convenience. In this paper we prove that if acd Q ∗ (G) < 9 / 2 or 0 < acd Q , even (G | N) < 4 , then G is solvable. Moreover, setting F ∈ { R , C } , we obtain the solvability of G by assuming acd F ∗ (G) < 29 / 8 or 0 < acd F , even (G | N) < 7 / 2 , and we conclude the solvability of N when 0 < acd F , even (G | N) < 18 / 5 . Replacing N by G in acd F , even (G | N) gives us an extended form of a result by Moreto and Nguyen. Examples are given to show that all the bounds are sharp. [ABSTRACT FROM AUTHOR]

Let G be a finite group, let Irr(G) be the set of all complex irreducible characters of G and let cd(G) be the set of all degrees of characters in [Formula: see text] Let [Formula: see text] be the set of all primes that divide some degrees in [Formula: see text] The character degree graph [Formula: see text] of G is the simple undirected graph with vertex set [Formula: see text] and in which two distinct vertices p and q are adjacent if there exists a character degree [Formula: see text] such that r is divisible by the product pq. In this paper we obtain a necessary condition for the character degree graph [Formula: see text] of a solvable group G to be eulerian. We also prove that every n – 2 regular graph on n vertices where n is even and [Formula: see text] is the character degree graph of some solvable group. In the last part of the paper we give a bound for the number of Eulerian character degree graphs in terms of number of vertices.

For a finite group G, a vanishing element of G is an element g ∈ G such that χ(g) = 0 for some irreducible complex character χ of G. Denote by πe(G) the set of orders of elements in G and by Vo(G) the set of the orders of vanishing elements of G. A group M is said to be recognizable if every finite group N with πe(M) = πe(N) is isomorphic to M, and V-recognizable if every finite group N with Vo(M) = Vo(N) is isomorphic to M. This paper investigates the relationship between recognizable and V-recognizable groups. As an application, we show that G ≅ M22 if and only if Vo(G) = Vo(M22). Hence we give a partial positive answer to [arXiv:1401.0300v12, Problem 19.30]. [ABSTRACT FROM AUTHOR]

In this paper, we study the determinant of irreducible representations of the generalized symmetric groups Z r ≀ S n . We give an explicit formula to compute the determinant of an irreducible representation of Z r ≀ S n . Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, given an integer n, r an odd prime and ζ a nontrivial multiplicative character of Z r ≀ S n with n < r, we obtain an explicit formula to compute N ζ (n) , the number of irreducible representations of Z r ≀ S n whose determinant is ζ. [ABSTRACT FROM AUTHOR]

G-character tables of a finite group G were defined in Felipe et al. (Quaest Math, 2022. ). These tables can be very useful to obtain certain structural information of a normal subgroup from the character table of G. We analyze certain structural properties of normal subgroups which can be determined using their G-character tables. For instance, we prove an extension of the Thompson’s theorem from minimal G-invariant characters of a normal subgroup. We also obtain a variation of Taketa’s theorem for hypercentral normal subgroups considering their minimal G-invariant characters. This generalization allows us to introduce a new class of nilpotent groups, the class of nMI-groups, whose members verify that its nilpotency class is bounded by the number of irreducible character degrees of the group. [ABSTRACT FROM AUTHOR]

In this paper, we study the determinant of irreducible representations of the generalized symmetric groups Z r ≀ S n . We give an explicit formula to compute the determinant of an irreducible representation of Z r ≀ S n . Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, given an integer n, r an odd prime and ζ a nontrivial multiplicative character of Z r ≀ S n with n < r, we obtain an explicit formula to compute N ζ (n) , the number of irreducible representations of Z r ≀ S n whose determinant is ζ. [ABSTRACT FROM AUTHOR]

We are given a finite group H, an automorphism τ of H of order r, a Galois extension L/K of fields of characteristic zero with cyclic Galois group ⟨ σ ⟩ of order r, and an absolutely irreducible representation ρ : H → GL (n , L) such that the action of τ on the character of ρ is the same as the action of σ . Then the following are equivalent. ∙ ρ is equivalent to a representation ρ ′ : H → GL (n , L) such that the action of σ on the entries of the matrices corresponds to the action of τ on H, and ∙ the induced representation ind H , H ⋊ ⟨ τ ⟩ (ρ) has Schur index one; that is, it is similar to a representation over K. As examples, we discuss a three dimensional irreducible representation of A 5 over Q [ 5 ] and a four dimensional irreducible representation of the double cover of A 7 over Q [ - 7 ] . [ABSTRACT FROM AUTHOR]

Let p be a prime and let G be a finite group such that the smallest prime that divides |G| is p. We find sharp bounds, depending on p, for the commuting probability and the average character degree to guarantee that G is nilpotent or supersolvable. [ABSTRACT FROM AUTHOR]

For an irreducible character χ of a finite group G, the codegree of χ is defined as ∣G: ker(χ)∣/χ(1). In this paper, the authors determine finite nonsolvable groups with exactly three nonlinear irreducible character codegrees, which are L2(2f) for f ≥ 2, PGL2(q) for odd q ≥ 5 or M10. [ABSTRACT FROM AUTHOR]