arXiv subjects
Publications and source records attributed to Maria Loukaki.
We completely characterize when three equal-order subgroups of a finite abelian group share a common transversal. Consequently, we determine when three full-rank lattices of equal volume with a discrete sum in $\mathbb{R}^d$ admit a common fundamental domain, answering, for $n=3$, an open question from 1997.
For any $n$ nonnegative integer a family of groups, denoted by $ \mathcal{D}_n $, was introduce by Bianchi et al., as the collection of all finite groups with exactly $n$ conjugacy classes of nontrivial, non self-normalizing subgroups. It was conjectured that $\mathcal{D}_5$ consists of solvable groups with derived length at most $3$. In this note we verify their conjecture.
Let $p$ be a prime number and $\zeta_p$ a primitive $p$-th root of unity. Chebotarev's theorem states that every square submatrix of the $p \times p$ matrix $(\zeta_p^{ij})_{i,j=0}^{p-1}$ is non-singular. In this paper we prove the same for principal submatrices of $(\zeta_n^{ij})_{i,j=0}^{n-1}$, when $n=pr$ is the product of two distinct primes, and $p$ is a large enough prime that has order $r-1$ in $\mathbf{Z}_r^*$. As an application, an uncertainty principle for cyclic groups of order $n$ is established when $n=pr$ as described above.
We investigate number-theoretic properties of the collection of nilpotent injectors or nilpotent projectors containing certain subgroups of finite soluble (or ${\mathcal N}$-constrained) groups.
Given a finite abelian group $G$ and cyclic subgroups $A$, $B$, $C$ of $G$ of the same order, we find necessary and sufficient conditions for $A$, $B$, $C$ to admit a common transversal for the cosets they afford. For an arbitrary number of cyclic subgroups we give a sufficient criterion when there exists a common complement. Moreover, in several cases where a common transversal exists, we provide concrete constructions.
Let $G$ be a $p$-group. We denote by $\mathcal{X}_i(G)$ the intersection of all subgroups of $G$ having index $p^i$, for $i \leq \log_p(|G|)$. In this paper, the newly introduced series $\{\mathcal{X}_i(G)\}_i$ is investigated and a number of results concerning its behaviour are proved. As an application of these results, we show that if an abelian subgroup $A$ of $G$ intersects each one of the subgroups $\mathcal{X}_i(G)$ at $\mathcal{X}_i(A)$, then $A$ has a complement in $G$. Conversely if an arbitrary subgroup $H$ of $G$ has a normal complement, then $\mathcal{X}_i(H) = \mathcal{X}_i(G) \cap H$.
An $n \times m$ non-negative matrix with row sum $m$ and column sum $n$ is called doubly stochastic. We answer the problem of finding doubly stochastic matrices of smallest posible support for every $1 <n \leq m$. Any matrix of minimum support is extremal in the sence of convexity, while examples of extremal matrices that are not of minimum support are given. But when $n,m$ are coprime integers extremal matrices are precisely those of minimum support.
Given a $p$-group $G$ and a subgroup-closed class $\mathfrak{X}$, we associate with each $\mathfrak{X}$-subgroup $H$ certain quantities which count $\mathfrak{X}$-subgroups containing $H$ subject to further properties. We show in Theorem I that each one of the said quantities is always $\equiv 1 \pmod p$ if and only if the same holds for the others. In Theorem II we supplement the above result by focusing on normal $\mathfrak{X}$-subgroups and in Theorem III we obtain a sharpened version of a celebrated theorem of Burnside relative to the class of abelian groups of bounded exponent. Various other corollaries are also presented.
Berkovich, Chillag and Herzog characterized all finite groups $G$ in which all the nonlinear irreducible characters of $G$ have distinct degrees. In this paper we extend this result showing that a similar characterization holds for all finite solvable groups $G$ that contain a normal subgroup $N$, such that all the irreducible characters of $G$ that do not contain $N$ in their kernel have distinct degrees.
Let $U_n$ denote the group of upper $n \times n$ unitriangular matrices over a fixed finite field $\mathbb{F}$ of order $q$. That is, $U_n$ consists of upper triangular $n \times n$ matrices having every diagonal entry equal to $1$. It is known that the degrees of all irreducible complex characters of $U_n$ are powers of $q$. It was conjectured by Lehrer that the number of irreducible characters of $U_n$ of degree $q^e$ is an integer polynomial in $q$ depending only on $e$ and $n$. We show that there exist recursive (for $n$) formulas that this number satisfies when $e$ is one of $1, 2$ and $3$, and thus show that the conjecture is true in those cases.
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.
Let $G$ be a finite $p$-solvable group, where $p$ is an odd prime. We establish a connection between extendible irreducible characters of subgroups of $G$ that lie under monomial characters of $G$ and nilpotent subgroups of $G$. We also provide a way to get ``good'' extendible irreducible characters inside subgroups of $G$. As an application, we show that every normal subgroup $N$ of a finite monomial odd $p, q$-group $G$, that has nilpotent length less than or equal to 3, is monomial.
Let $G$ be a finite group of odd order, $\F$ a finite field of odd characteristic $p$ and $\B$ a finite--dimensional symplectic $\F G$-module. We show that $\B$ is $\F G$-hyperbolic, i.e., it contains a self--perpendicular $\F G$-submodule, iff it is $\F N$-hyperbolic for every cyclic subgroup $N$ of $G$.
Nearly twenty years ago Isaacs and the first author of this paper wrote a series of articles \cite{isa2}, \cite{da3}, \cite{da2} about what were called ``stabilizer limits'' of group characters, following the terminology of Berger \cite{be}. The second author, in her thesis \cite{lo}, needed one of the results of those articles in a new situation which was not treated earlier. Eventually she was able, by complicated and delicate arguments, to reduce her proof to a special case where \cite[Theorem 8.4]{da3} could be applied. But this approach was extremely awkward. In the present paper we use arguments similar to those in the earlier articles to prove a Main Theorem from which the exact Theorem A needed in \cite{lo} easily follows.
A finite group $G$ is called monomial if every irreducible character of $G$ is induced from a linear character of some subgroup of $G$. One of the main questions regarding monomial groups is whether or not a normal subgroup $N$ of a monomial group $G$ is itself monomial. In the case that $G$ is a group of even order, it has been proved (Dade, van der Waall) that $N$ need not be monomial. Here we show that, if $G$ is a monomial group of order $p^aq^b$, where $p$ and $q$ are distinct odd primes, then any normal subgroup $N$ of $G$ is also monomial.