Set Of Irreducible Character Degrees Research Articles

Character degrees of 5-groups of maximal class

Abstract Let 𝐺 be a 5-group of maximal class with major centralizer G 1 = C G ⁢ ( G 2 / G 4 ) G_{1}=C_{G}({G_{2}}/{G_{4}}) . In this paper, we prove that the irreducible character degrees of a 5-group 𝐺 of maximal class are almost determined by the irreducible character degrees of the major centralizer G 1 G_{1} and show that the set of irreducible character degrees of a 5-group of maximal class is either { 1 , 5 , 5 3 } \{1,5,5^{3}\} or { 1 , 5 , … , 5 k } \{1,5,\ldots,5^{k}\} with k ≥ 1 k\geq 1 .

An overview on the bipartite divisor graph for the set of irreducible character degrees

Let G be a finite group. The bipartite divisor graph B(G) for the set of irreducible complex character degrees cd(G) is the undirected graph with vertex set consisting of the prime numbers dividing some element of cd(G) and of the nonidentity character degrees in cd(G), where a prime number p is declared to be adjacent to a character degree m if and only if p divides m. The graph B(G) is bipartite and it encodes two of the most widely studied graphs associated to the character degrees of a finite group: the prime graph and the divisor graph on the set of irreducible character degrees. The scope of this paper is two-fold. We draw some attention to B(G) by outlining the main results that have been proved so far, see for instance [10; 11; 25; 26; 27]. In this process we improve some of these results.

Groups where the centers of the irreducible characters form a chain

We consider groups where the centers of the irreducible characters form a chain. We obtain two alternate characterizations of these groups, and we obtain some information regarding the structure of these groups. Using our results, we are able to classify those groups where the kernels of the irreducible characters form a chain. We show that a result of Nenciu regarding nested GVZ groups is really a result about nested groups. We obtain an alternate proof of a theorem of Isaacs regarding the existence of p-groups with a given set of irreducible character degrees.

Some Nonsolvable Character Degree Sets

For positive integer k and nonabelian simple group S, let $$S^{k}$$ be the direct product of k copies of S. We conjecture that all finite groups G with $$\mathrm{cd}(G)=\mathrm{cd}(S^{k})$$ are quasi perfect groups (that is; $$G'=G''$$) and hence nonsolvable groups, where $$\mathrm{cd}(G)$$ is the set of irreducible character degrees of G. In this paper, we prove this conjecture for $$S\in \{\mathrm{PSL}_{2}(p^{f}), \mathrm{PSL}_{2}(2^{f}), \mathrm{Sz}(q)\}$$, where $$p>2$$ is an odd prime number such that $$p^{f}>5$$ and $$p^{f}\pm 1\not \mid 2^{k}$$, and $$q=2^{2n+1}\geqslant 8$$.

Isaacs–Seitz conjecture for certain groups

Let G be a finite group. I. M. Isaacs and G. Seitz have conjectured that if G is a solvable group, then Taketa’s inequality holds, where is the set of irreducible character degrees of G and is the derived length of G. In this paper, we show that this inequality holds if G is a solvable Frobenius group. Also, we prove that if for some normal Sylow p-subgroup P of G, then . Moreover, we investigate this conjecture for some sets of the real-valued irreducible character degrees.

On regular bipartite divisor graph for the set of irreducible character degrees

Given a finite group $G$, the \textit{bipartite divisor graph}, denoted by $B(G)$, for its irreducible character degrees is the bipartite graph with bipartition consisting of $cd(G)^{*}$, where $cd(G)^{*}$ denotes the nonidentity irreducible character degrees of $G$ and the $\rho(G)$ which is the set of prime numbers that divide these degrees, and with $\{p,n\}$ being an edge if $\gcd(p,n)\neq 1$. In [Bipartite divisor graph for the set of irreducible character degress, Int. J. Group Theory, 2017], the author considered the cases where $B(G)$ is a path or a cycle and discussed some properties of $G$. In particular she proved that $B(G)$ is a cycle if and only if $G$ is solvable and $B(G)$ is either a cycle of length four or six. Inspired by $2$-regularity of cycles, in this paper we consider the case where $B(G)$ is an $n$-regular graph for $n\in\{1,2,3\}$. In particular we prove that there is no solvable group whose bipartite divisor graph is $C_{4}+C_{6}$.

The number of composite character degrees in solvable groups

ABSTRACTLet cd(G) be the set of irreducible character degrees of a finite group G. Also, let cdP(G)⊆cd(G) and cdC(G)⊆cd(G) be the sets of all prime character degrees and all composite character degrees of G, respectively. We prove that for a solvable group G, if |cdC(G)|≥4, then . This extends the results of [4], [7], [2] and [3] where it has been proved that |cdP(G)|≤3 if |cdC(G)| = 0,1,or2, and |cdP(G)|≤4 if |cdC(G)| = 3, respectively.

Bipartite divisor graph for the set of irreducible character degrees

‎Let $G$ be a finite group‎. ‎We consider the set of the irreducible complex characters of $G$‎, ‎namely $Irr(G)$‎, ‎and the related degree set $cd(G)={chi(1)‎ : ‎chiin Irr(G)}$‎. ‎Let $rho(G)$ be the set of all primes which divide some character degree of $G$‎. ‎In this paper we introduce the bipartite divisor graph for $cd(G)$ as an undirected bipartite graph with vertex set $rho(G)cup (cd(G)setminus{1})$‎, ‎such that an element $p$ of $rho(G)$ is adjacent to an element $m$ of $cd(G)setminus{1}$ if and only if $p$ divides $m$‎. ‎We denote this graph simply by $B(G)$‎. ‎Then by means of combinatorial properties of this graph‎, ‎we discuss the structure of the group $G$‎. ‎In particular‎, ‎we consider the cases where $B(G)$ is a path or a cycle‎.

Extending Huppert’s conjecture from non-Abelian simple groups to quasi-simple groups

We propose to extend a conjecture of Bertram Huppert [Illinois J. Math. 44 (2000) 828–842] from finite non-Abelian simple groups to finite quasi-simple groups. Specifically, we conjecture that if a finite group $G$ and a finite quasi-simple group $H$ with ${\mathrm{Mult}}(H/\mathbf{Z}(H))$ cyclic have the same set of irreducible character degrees (not counting multiplicity), then $G$ is isomorphic to a central product of $H$ and an Abelian group. We present a pattern to approach this extended conjecture and, as a demonstration, we confirm it for the special linear groups in dimensions $2$ and $3$.

Recognition of $$\text {PSL}(2,p)$$ PSL ( 2 , p ) by order and some information on its character degrees where $$p$$ p is a prime

Let \(G\) be a finite group and \(\text {cd}(G)\) be the set of irreducible character degrees of \(G\). In this paper we prove that if \(p\) is a prime number, then the simple group \(\text {PSL}(2,p)\) is uniquely determined by its order and some information about its character degrees. In fact we prove that if \(G\) is a finite group such that (i) \(|G|=|\text {PSL}(2,p)|\), (ii) \(p\in \text {cd}(G)\), (iii) \(\text {cd}(G)\) has an even integer, and (iv) there does not exist any element \(a\in \text {cd}(G)\) such that \(2p\mid a\), then \(G\cong \text {PSL}(2,p)\). As a consequence of our result we get that \(\text {PSL}(2,p)\) is uniquely determined by its order and the largest and the second largest character degrees.

ON HUPPERT’S CONJECTURE FOR THE CONWAY AND FISCHER FAMILIES OF SPORADIC SIMPLE GROUPS

AbstractLet $G$ denote a finite group and $\mathrm{cd} (G)$ the set of irreducible character degrees of $G$. Huppert conjectured that if $H$ is a finite nonabelian simple group such that $\mathrm{cd} (G)= \mathrm{cd} (H)$, then $G\cong H\times A$, where $A$ is an abelian group. He verified the conjecture for many of the sporadic simple groups and we complete its verification for the remainder.

Nonsolvable normal subgroups and irreducible character degrees

Let the nonsolvable N be a normal subgroup of the finite group G and cd(G|N) denote the irreducible character degrees of G such that there exist respectively corresponding character kernels not containing N. Write |cd(G|N)| to stand for the cardinality of cd(G|N). Suppose that |cd(G|N)|⩽5. In this paper, we prove that if N is a minimal normal subgroup, then N is a simple group of Lie type. When N is a normal subgroup, we prove that G has a normal series 1⩽V<U⩽N⩽G such that V is solvable, U/V is a simple group of Lie type and the cardinality |cd(G/U)|⩽3. As an application, we investigate the structure of G when 5⩽|cd(G)|⩽6. Here cd(G) denotes the set of irreducible character degrees of G.

Huppert's Conjecture for $Fi_{23}$

Let $G$ denote a finite group and $\text{cd}(G)$ the set of irreducible character degrees of $G$. Bertram Huppert conjectured that if $H$ is a finite nonabelian simple group such that $\text{cd}(G) =\text{cd}(H)$, then $G\cong H \times A$, where $A$ is an abelian group. Huppert verified the conjecture for many of the sporadic simple groups. We illustrate the arguments by presenting the verification of Huppert's Conjecture for $Fi\_{23}$.

Verifying Huppert’s Conjecture for PSp4(q) when q &gt; 7

Let G denote a finite group and cd (G) the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd (G) = cd (H), then G ≅ H × A, where A is an abelian group. Huppert verified the conjecture for PSp4(q) when q = 3, 4, 5, or 7. In this paper, we extend Huppert’s results and verify the conjecture for PSp4(q) for all q. This demonstrates progress toward the goal of verifying the conjecture for all nonabelian simple groups of Lie type of rank two.

Verifying Huppert’s Conjecture for 2 G 2(q 2)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups 2G2(q2) for q2 = 32m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many of the nonabelian simple groups.

Degree Graphs of Simple Groups

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph ∆(G) is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ∈ cd(G). We compile here the graphs ∆(G) for all finite simple groups G.

Verifying Huppert's Conjecture for PSL3(q) and PSU3(q 2)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H × A, where A is an abelian group. We examine arguments to verify this conjecture for the simple groups of Lie type of rank two. To illustrate our arguments, we extend Huppert's results and verify the conjecture for the simple linear and unitary groups of rank two.

Diameters of degree graphs of nonsolvable groups, II

Let G be a finite group and let cd ( G ) be the set of irreducible character degrees of G. The degree graph Δ ( G ) is the graph whose set of vertices is the set of primes that divide degrees in cd ( G ) , with an edge between p and q if pq divides a for some degree a ∈ cd ( G ) . It is shown using the degree graphs of the finite simple groups that if G is a nonsolvable group, then the diameter of Δ ( G ) is at most 3.

Degree Graphs of Simple Linear and Unitary Groups

Let G be a finite group and let cd (G) be the set of irreducible character degrees of G. The degree graph Δ(G) is the graph whose set of vertices is the set of primes that divide degrees in cd (G), with an edge between p and q if pq divides a for some degree a ∈ cd (G). We determine the graph Δ(G) for the finite simple groups of types A ℓ(q) and 2 A ℓ (q 2), that is, for the simple linear and unitary groups.

Degree Graphs of Simple Groups of Exceptional Lie Type#

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph Δ(G) of G is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ∈ cd(G). In this paper, we determine the graph Δ(G) when G is a finite simple group of exceptional Lie type.