Let $G$ be a finite group and $nse\left(G\right)$ the set of numbers of elements with the same order in $G$. In this paper, we prove that a finite group $G$ is isomorphic to $M$, where $M$ is one of the Mathieu groups, if and only if the following hold: (1) $\left|G\right|=\left|M\right|$, (2) $nse\left(G\right)=nse\left(M\right)$.

In recent years, starting with the paper [B-D-S], we have investigated the possibility of characterizing countable subgroups of the torus $T=\mathbf{R}/\mathbf{Z}$ by subsets of $\mathbf{Z}$. Here we consider new types of subgroups: let $K\subseteq T$ be a Kronecker set (a compact set on which every continuous function $f:K\to T$ can be uniformly approximated by characters of $T$), and $G$ the group generated by $K$. We prove (Theorem 1) that $G$ can be characterized by a subset of ${\mathbf{Z}}^2$ (instead of a subset of $\mathbf{Z}$). If $K$ is finite, Theorem 1 implies our...

1. Introduction. Let p be a prime number and ${\mathbb{Z}}_p$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, $k_{\infty }$ a ${\mathbb{Z}}_p$-extension of k, $k_n$ the nth layer of $k_{\infty }/k$, and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$. Iwasawa proved the following well-known theorem about the order $A_n$ of $A_n$: Theorem A (Iwasawa). Let $k_{\infty }/k$ be a ${\mathbb{Z}}_p$-extension and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$, where $k_n$ is the $n$th layer of $k_{\infty }/k$. Then there exist integers $\lambda =\lambda (k_{\infty }/k)\ge 0$, $\mu =\mu (k_{\infty }/k)\ge 0$, $\nu =\nu (k_{\infty }/k)$, and n₀ ≥ 0...

We present several results related to $L\Sigma \left(\kappa \right)$-spaces where $\kappa$ is a finite cardinal or $\omega$; we consider products and some constructions that lead from spaces of these classes to other spaces of similar classes.

Let $L\left(X\right)$ be the space of all lower semi-continuous extended real-valued functions on a Hausdorff space $X$, where, by identifying each $f$ with the epi-graph $epi\left(f\right)$, $L\left(X\right)$ is regarded the subspace of the space ${Cld}_F^{*}(X\times ℝ)$ of all closed sets in $X\times ℝ$ with the Fell topology. Let $$LSC\left(X\right)=\{f\in L\left(X\right)\mid f\left(X\right)\cap ℝ\ne \varnothing ,f\left(X\right)\subset (-\infty ,\infty ]\}\text{and}{LSC}_B\left(X\right)=\{f\in L\left(X\right)\mid f\left(X\right)\text{is}\text{a}\text{bounded}\text{subset}\text{of}ℝ\}.$$ We show that $L\left(X\right)$ is homeomorphic to the Hilbert cube $Q={[-1,1]}^{\mathbb{N}}$ if and only if $X$ is second countable, locally compact and infinite. In this case, it is proved that $(L\left(X\right),LSC\left(X\right),{LSC}_B\left(X\right))$ is homeomorphic to $(ConeQ,Q\times (0,1),\Sigma \times (0,1\left)\right)$ (resp. $(Q,s,\Sigma )$) if $X$ is compact (resp. $X$ is non-compact), where $ConeQ=(Q\times 𝐈)/(Q\times \{1\left\}\right)$ is...