For a complex character $\chi$ of a finite group $G$, it is known that the product $\mathrm{sh}\left(\chi \right)={\prod }_{l\in L\left(\chi \right)}(\chi \left(1\right)-l)$ is a multiple of $\left|G\right|$, where $L\left(\chi \right)$ is the image of $\chi$ on $G-\left\{1\right\}$. The character $\chi$ is said to be a sharp character of type $L$ if $L=L\left(\chi \right)$ and $\mathrm{sh}\left(\chi \right)=\left|G\right|$. If the principal character of $G$ is not an irreducible constituent of $\chi$, then the character $\chi$ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups $G$ with normalized sharp characters of type $\{-1,0,2\}$. Here we prove that such a group with nontrivial...

Let $V$ be the complex vector space of homogeneous linear polynomials in the variables $x_1,...,x_m$. Suppose $G$ is a subgroup of $S_m$, and $\chi$ is an irreducible character of $G$. Let $H_d(G,\chi )$ be the symmetry class of polynomials of degree $d$ with respect to $G$ and $\chi$. For any linear operator $T$ acting on $V$, there is a (unique) induced operator $K_{\chi }\left(T\right)\in \mathrm{End}\left(H_d(G,\chi )\right)$ acting on symmetrized decomposable polynomials by $$K_{\chi }\left(T\right)(f_1*f_2*...*f_d)=T{f}_1*T{f}_2*...*T{f}_d.$$ In this paper, we show that the representation $T\mapsto K_{\chi }\left(T\right)$ of the general linear group $GL\left(V\right)$ is equivalent to the direct sum of $\chi \left(1\right)$ copies...

Let $G$ be a finite group and $p$ a prime number. We prove that if $G$ is a finite group of order $\left|\mathrm{PSL}\right(2,p^2\left)\right|$ such that $G$ has an irreducible character of degree $p^2$ and we know that $G$ has no irreducible character $\theta$ such that $2p\mid \theta \left(1\right)$, then $G$ is isomorphic to $\mathrm{PSL}(2,p^2)$. As a consequence of our result we prove that $\mathrm{PSL}(2,p^2)$ is uniquely determined by the structure of its complex group algebra.

Let $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there...

Let $N$ be a normal subgroup of a group $G$. The structure of $N$ is given when the $G$-conjugacy class sizes of $N$ is a set of a special kind. In fact, we give the structure of a normal subgroup $N$ under the assumption that the set of $G$-conjugacy class sizes of $N$ is $(p_{1n_1}^{a_{1n_1}},\cdots ,p_{11}^{a_{11}},1)\times \cdots \times (p_{r{n}_r}^{a_{r{n}_r}},\cdots ,p_{r1}^{a_{r1}},1)$, where $r>1$, $n_i>1$ and $p_{ij}$ are distinct primes for $i\in \{1,2,\cdots ,r\}$, $j\in \{1,2,\cdots ,n_i\}$.