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Edith Adan-Bante

Publications and source records attributed to Edith Adan-Bante.

At least 19 recordsLinked to original sources
arXiv2009

On conjugacy classes and derived length

Let $G$ be a finite group and $A$, $B$ and $D$ be conjugacy classes of $ G$ with $D\subseteq AB=\{xy\mid x\in A, y\in B\}$. Denote by $η(AB)$ the number of distinct conjugacy classes such that $AB$ is the union of those. Set ${\bf C}_G(A)=\{g\in G\mid x^g=x {for all} x\in A\}$. If $AB=D$ then ${\bf C}_G(D)/({\bf C}_G(A)\cap{\bf C}_G(B))$ is an abelian group. If, in addition, $G$ is supersolvable, then the derived length of ${\bf C}_G(D)/({\bf C}_G(A)\cap{\bf C}_G(B))$ is bounded above by $2η(AB)$.

math.GR↗
arXiv2009

On conjugacy classes of SL$(2,q)$

Let SL(2,q) be the group of 2X2 matrices with determinant one over a finite field F of size q. We prove that if q is even, then the product of any two noncentral conjugacy classes of SL(2,q) is the union of at least q-1 distinct conjugacy classes of SL(2,q). On the other hand, if q>3 is odd, then the product of any two noncentral conjugacy classes of SL(2,q) is the union of at least (q+3)/2 distinct conjugacy classes of SL(2,q).

math.GR↗
arXiv2009

On conjugacy classes of GL(n,q) and SL(n,q)

Let GL(n,q) be the group of nxn invertible matrices over a field with q elements, and SL(n,q) be the group of nxn matrices with determinant 1 over a field with q elements. We prove that the product of any two non-central conjugacy classes in GL(n,q) is the union of at least q-1 distinct conjugacy classes, and that the product of any two non-central conjugacy classes in SL(n,q) is the union of at least $\lceil\frac{q}{2} \rceil$ distinct conjugacy classes.

math.GR↗
arXiv2008

Characters of prime degree

Let $G$ be a finite nilpotent group, $χ$ and $ψ$ be irreducible complex characters of $G$ of prime degree. Assume that $χ(1)=p$. Then either the product $χψ$ is a multiple of an irreducible character or $χψ$ is the linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters.

math.GR↗
arXiv2007

Symmetric groups and conjugacy classes

Let S_n be the symmetric group on n-letters. Fix n>5. Given any nontrivial $α,β\in S_n$, we prove that the product $α^{S_n}β^{S_n}$ of the conjugacy classes $α^{S_n}$ and $β^{S_n}$ is never a conjugacy class. Furthermore, if n is not even and $n$ is not a multiple of three, then $α^{S_n}β^{S_n}$ is the union of at least three distinct conjugacy classes. We also describe the elements $α,β\in S_n$ in the case when $α^{S_n}β^{S_n}$ is the union of exactly two distinct conjugacy classes.

math.GR↗
arXiv2007

Restriction of characters and products of characters

Let G be a finite p-group, for some prime p, and $ψ, θ\in \Irr(G)$ be irreducible complex characters of G. It has been proved that if, in addition, $ψ,θ$ are faithful characters, then the product $ψθ$ is a multiple of an irreducible or it is the nontrivial linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters of G. We show that if we do not require the characters to be faithful, then given any integer k>0, we can always find a p-group G and irreducible characters $Ψ$ and $Θ$ such that $ΨΘ$ is the nontrivial combination of exactly k distinct irreducible characters. We do this by translating examples of decompositions of restrictions of characters into decompositions of products of characters.

math.GR↗
arXiv2006

Homogeneous products of conjugacy classes

Let $G$ be a finite group and $a\in G$. Let $a^G=\{g^{-1}ag\mid g\in G\}$ be the conjugacy class of $a$ in $G$. Assume that $a^G$ and $b^G$ are conjugacy classes of $G$ with the property that ${\bf C}_G(a)={\bf C}_G(b)$. Then $a^G b^G$ is a conjugacy class if and only if $[a,G]=[b,G]=[ab,G]$ and $[ab,G]$ is a normal subgroup of $G$.

math.GR↗
arXiv2006

Derived Length and Products of Conjugacy Classes

Let $G$ be a supersolvable group and $A$ be a conjugacy class of $G$. Observe that for some integer $η(AA^{-1})>0$, $AA^{-1}=\{a b^{-1}\mid a,b\in A\}$ is the union of $η(AA^{-1})$ distinct conjugacy classes of $G$. Set ${\bf C}_G(A)=\{g\in G\mid a^g=a\text{for all} a\in A\}$. Then the derived length of $G/{\bf C}_G(A)$ is less or equal than $2η(A A^{-1})-1$.

math.GR↗
arXiv2005

Induction of Characters and Finite $p$-Groups

Let $G$ be a finite $p$-group, where $p$ is an odd prime number, $H$ be a subgroup of $G$ and $θ\in \Irr(H)$ be an irreducible character of $H$. Assume also that $|G:H|=p^2$. Then the character $θ^G$ of $ G$ induced by $θ$ is either a multiple of an irreducible character of $G$, or has at least $\frac{p+1}{2}$ distinct irreducible constituents.

math.GR↗
arXiv2005

Squares of characters and finite groups

Let $G$ be a group of odd order and $χ$ be a complex irreducible character. Then there exists a unique character $χ^{(2)}\in\Irr(G)$ such that $[χ^2,χ^{(2)}]$ is odd. Also, there exists a unique character $ψ\in \Irr(G)$ such that $[ψ^2, χ]$ is odd.

math.GR↗
arXiv2005

On nilpotent groups and conjugacy classes

Let $G$ be a nilpotent group and $a\in G$. Let $a^G=\{g^{-1}ag\mid g\in G\}$ be the conjugacy class of $a$ in $G$. Assume that $a^G$ and $b^G$ are conjugacy classes of $G$ with the property that $|a^G|=|b^G|=p$, where $p$ is an odd prime number. Set $a^G b^G=\{xy\mid x\in a^G, y\in b^G\}$. Then either $a^G b^G=(ab)^G$ or $a^G b^G$ is the union of at least $\frac{p+1}{2}$ distinct conjugacy classes. As an application of the previous result, given any nilpotent group $G$ and any conjugacy class $a^G$ of size $p$, we describe the square $a^G a^G$ of $a^G$ in terms of conjugacy classes of $G$.

math.GR↗
arXiv2005

Conjugacy classes and finite $p$-groups

Let $G$ be a finite $p$-group, where $p$ is a prime number, and $a\in G$. Denote by $\Cl(a)=\{gag^{-1}\mid g\in G\}$ the conjugacy class of $a$ in $G$. Assume that $|\Cl(a)|=p^n$. Then $\Cl(a)\Cl(a^{-1})=\{xy\mid x\in \Cl(a), y\in \Cl(a^{-1})\}$ is the union of at least $n(p-1)+1$ distinct conjugacy classes of $G$.

math.GR↗
arXiv2004

Homogeneous products of characters

I. M. Isaacs has conjectured (see \cite{isa00}) that if the product of two faithful irreducible characters of a solvable group is irreducible, then the group is cyclic. In this paper we prove a special case of the following conjecture, which generalizes Isaacs conjecture. Suppose that $G$ is solvable and that $ψ,ϕ\in\Irr(G)$ are faithful. If $ψϕ=mχ$ where $m$ is a positive integer and $χ\in \Irr(G)$ then $ψ$ and $ϕ$ vanish on $G- Z(G)$. In particular we prove that the above conjecture holds for $p$-groups.

math.GR↗
arXiv2004

Products of characters and derived length II

Let $G$ be a finite solvable group and $χ, ψ\in \Irr(G)$ be complex characters of $G$. Let $α$ be an irreducible constituent of the product $χψ$. We show that the derived length of $\Ker(α)/\Ker(χψ)$ is bounded by a linear function on the number of distinct irreducible constituents of $χψ$

math.GR↗
arXiv2003

Products of characters and finite p-groups

Let G be a finite p-group, where p is a prime number, and $χ$ and $ψ$ be faithful complex irreducible characters of G. We study the relation between the number $η(χ,ψ)$ of distinct irreducible constituents of the product $χψ$ and p.

math.GR↗