• Volume 114, Issue 3

August 2004,   pages  217-298

• On Finite Groups whose Every Proper Normal Subgroup is a Union of a Given Number of Conjugacy Classes

Let 𝐺 be a finite group and 𝐴 be a normal subgroup of 𝐺. We denote by $ncc(A)$ the number of 𝐺-conjugacy classes of 𝐴 and 𝐴 is called 𝑛-decomposable, if $ncc(A)=n$. Set $\mathcal{K}_G=\{ncc(A)|A\vartriangleleft G\}$. Let 𝑋 be a non-empty subset of positive integers. A group 𝐺 is called 𝑋-decomposable, if $\mathcal{K}_G=X$.

Ashrafi and his co-authors [1–5] have characterized the 𝑋-decomposable non-perfect finite groups for $X=\{1,n\}$ and 𝑛 ≤ 10. In this paper, we continue this problem and investigate the structure of 𝑋-decomposable non-perfect finite groups, for $X=\{1, 2, 3\}$. We prove that such a group is isomorphic to $Z_6, D_8, Q_8, S_4$, Small Group (20,3), Small Group (24,3), where Small Group (𝑚, 𝑛) denotes the $m^{\mathrm{th}}$ group of order 𝑛 in the small group library of GAP [11].

• Characteristic Properties of Large Subgroups in Primary Abelian Groups

Suppose 𝐺 is an arbitrary additively written primary abelian group with a fixed large subgroup 𝐿. It is shown that 𝐺 is (a) summable; (b)$\sum$-summable; (c) a $\sum$-group; (d) $p^{𝜔+1}$-projective only when so is 𝐿. These claims extend results of such a kind obtained by Benabdallah, Eisenstadt, Irwin and Poluianov, Acta Math. Acad. Sci. Hungaricae (1970) and Khan, Proc. Indian Acad. Sci. Sect. A (1978).

• Multilinear Integral Operators and Mean Oscillation

In this paper, the boundedness properties for some multilinear operators related to certain integral operators from Lebesgue spaces to Orlicz spaces are obtained. The operators include Calderón–Zygmund singular integral operator, fractional integral operator, Littlewood–Paley operator and Marcinkiewicz operator.

• Superstability of the Generalized Orthogonality Equation on Restricted Domains

Chmieliński has proved in the paper [4] the superstability of the generalized orthogonality equation $|\langle f(x),f(y)\rangle|=|\langle x,y\rangle|$. In this paper, we will extend the result of Chmieliński by proving a theorem: Let $D_n$ be a suitable subset of $\mathbb{R}^n$. If a function $f: D_n→\mathbb{R}^n$ satisfies the inequality $||\langle f(x),f(y)\rangle|-|\langle x,y\rangle||≤\varphi(x,y)$ for an appropriate control function $\varphi(x,y)$ and for all $x,y \in D_n$, then 𝑓 satisfies the generalized orthogonality equation for any $x,y \in D_n$.

• Non-Linear Second-Order Periodic Systems with Non-Smooth Potential

In this paper we study second order non-linear periodic systems driven by the ordinary vector 𝑝-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth conditions on the potential function. Then we establish the existence of non-trivial homoclinic (to zero) solutions. Our theorem appears to be the first such result (even for smooth problems) for systems monitored by the 𝑝-Laplacian. In the last section of the paper we examine the scalar non-linear and semilinear problem. Our approach uses a generalized Landesman–Lazer type condition which generalizes previous ones used in the literature. Also for the semilinear case the problem is at resonance at any eigenvalue.

• Proceedings – Mathematical Sciences

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