scholarly journals On projective characters of prime degree

1991 ◽  
Vol 33 (3) ◽  
pp. 311-321 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Prime Number ◽  
Structural Information ◽  
Alternating Group ◽  
Complex Numbers ◽  
Prime Degree ◽  
Counter Example ◽  
Projective Representations

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results.Let Proj(G, α) denote the set of irreducible projective characters of a group G with cocycle α. In a previous paper [3] the author showed that if G is a (p, α)-group, that is the degrees of the elements of Proj(G, α) are all powers of a prime number p, then G is solvable. However Isaacs and Passman in [8] were able to give structural information about a group G for which ξ(1) divides pe for all ξ ∈ Proj(G, 1), where 1 denotes the trivial cocycle of G, and indeed classified all such groups in the case e = l. Their results rely on the fact that G has a normal abelian p-complement, which is false in general if G is a (p, α)-group; the alternating group A4 providing an easy counter-example for p = 2.

scholarly journals On projective characters of the same degree

1998 ◽  
Vol 40 (3) ◽  
pp. 431-434 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Complex Numbers ◽  
Central Type ◽  
Projective Representations

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results. Let Proj (G, α) denote the set of irreducible projective characters of a group G with cocyle α. In previous papers (for exampe [2], [4], and [6]) numerous authors have considered the situation when Proj(G, α) = 1 or 2; such groups are said to be of α-central type or of 2α-central type, respectively. In particular in [4, Theorem A] the author showed that if Proj(G, α) = {ξ1, ξ2}, then ξ1(1)=ξ2(1). This result has recently been independently confirmed in [8, Corollary C].

A counter-example in summability theory

1978 ◽  
Vol 83 (3) ◽  
pp. 353-355
Author(s):  
B. Kuttner
Keyword(s):  
Summability Method ◽  
Complex Numbers ◽  
Tauberian Condition ◽  
Counter Example ◽  
Image Position ◽  
Regular Summability ◽  
Natural Way

Let Σ denote the set of all seriesof complex numbers. By a ‘summability method’, say A, we mean a function from some subset (the set of ‘A -summable series’) of Σ into the set of complex numbers. We will use the language generally associated with this definition, and will take for granted the case in which A is (C, 1). A summability method A will be called linear if, whenever a, b are A -summable, then so is λa + μb (where λ, μ are any complex constants) and if the. A -sums of a, b, λa + μb are then related in the natural way. We call A regular if, whenever a converges to σ, it is A -summable to σ. If A is a regular summability method, then any condition P on the series (1) will be called a Tauberian condition for A if any A -summable series which satisfies P is convergent.

Congruence of Ankeny–Artin–Chowla type for cyclic fields of prime degree l

1996 ◽  
Vol 119 (1) ◽  
pp. 17-22 ◽  
Author(s):  
Stanislav Jakubec
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Quadratic Field ◽  
Fundamental Unit ◽  
Prime Degree ◽  
Image Position

Ankeny–Artin–Chowla obtained several congruences for the class number hk of a quadratic field K, some of which were also obtained by Kiselev. In particular, if the discriminant of K is a prime number p ≡ 1 (mod 4) and ε = t + u √p/2 is the fundamental unit of K, then

Groups which act transitively on the projective characters of a normal subgroup

1988 ◽  
Vol 104 (3) ◽  
pp. 429-434 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Normal Subgroup ◽  
Complex Numbers ◽  
Projective Representations

All groups G considered in this paper are finite and all representations of G are defined over the complex numbers. The reader unfamiliar with projective representations is referred to [4] for basic definitions and elementary results.

PROJECTIVE CHARACTERS WITH PRIME POWER DEGREES

2018 ◽  
Vol 99 (1) ◽  
pp. 78-82
Author(s):  
YANG LIU
Keyword(s):  
Structural Information ◽  
Character Degrees ◽  
Character Theory ◽  
Induced Representations ◽  
Projective Character ◽  
Projective Representations ◽  
A Finite Group ◽  
The Relationship ◽  
Factor Set ◽  
Representation Group

We consider the relationship between structural information of a finite group $G$ and $\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$, the set of all irreducible projective character degrees of $G$ with factor set $\unicode[STIX]{x1D6FC}$. We show that for nontrivial $\unicode[STIX]{x1D6FC}$, if all numbers in $\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$ are prime powers, then $G$ is solvable. Our result is proved by classical character theory using the bijection between irreducible projective representations and irreducible constituents of induced representations in its representation group.

Groups whose projective characters are Galois conjugate

1989 ◽  
Vol 106 (2) ◽  
pp. 193-197 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Complex Numbers ◽  
Projective Representations

All groups G considered in this paper are finite and all representations of G are defined over the complex numbers. The reader unfamiliar with projective representations is referred to [6] for basic definitions and elementary results.

scholarly journals ALTERNATING AND SYMMETRIC GROUPS WITH EULERIAN GENERATING GRAPH

2017 ◽  
Vol 5 ◽  
Author(s):  
ANDREA LUCCHINI ◽  
CLAUDE MARION
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Prime Number ◽  
Symmetric Groups ◽  
Alternating Group ◽  
Generating Graph ◽  
Alternating And Symmetric Groups ◽  
A Finite Group

Given a finite group $G$, the generating graph $\unicode[STIX]{x1D6E4}(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we investigate properties about the degrees of the vertices of $\unicode[STIX]{x1D6E4}(G)$ when $G$ is an alternating group or a symmetric group of degree $n$. In particular, we determine the vertices of $\unicode[STIX]{x1D6E4}(G)$ having even degree and show that $\unicode[STIX]{x1D6E4}(G)$ is Eulerian if and only if $n\geqslant 3$ and $n$ and $n-1$ are not equal to a prime number congruent to 3 modulo 4.

Groups with two projective characters

1988 ◽  
Vol 103 (1) ◽  
pp. 5-14 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Complex Numbers ◽  
Projective Representations

All representations and characters studied in this paper are taken over the field of complex numbers, and all groups considered are finite. The reader unfamiliar with projective representations is referred to [8] for basic definitions and terminology.

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