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Let $sigma={sigma_{i}|iin I}$ be some partition of the set $mathbb{P}$ of all primes, that is, $mathbb{P}=bigcup_{iin I}sigma_{i}$ and $sigma_{i}cap sigma_{j}=emptyset$ for all $i eq j$. Let $G$ be a finite group. A set $mathcal {H}$ of subgroups of $G$ is said to be a complete Hall $sigma$-set of $G$ if every non-identity member of $mathcal {H}$ is a Hall $sigma_{i}$-subgroup of $G$ and $mathcal {H}$ contains exactly one Hall $sigma_{i}$-subgroup of $G$ for every $sigma_{i}in sigma(G)$. $G$ is said to be a $sigma$-group if it possesses a complete Hall $sigma$-set. A $sigma$-group $G$ is said to be $sigma$-dispersive provided $G$ has a normal series $1 = G_1<G_2<cdots< G_t< G_{t+1} = G$ and a complete Hall $sigma$-set ${H_{1}, H_{2}, cdots, H_{t}}$ such that $G_iH_i = G_{i+1}$ for all $i= 1,2,ldots t$. In this paper, we give a characterizations of $sigma$-dispersive group, which give a positive answer to an open problem of Skiba in the paper.
We consider sparse representations of signals from redundant dictionaries which are unions of several orthonormal bases. The spark introduced by Donoho and Elad plays an important role in sparse representations. However, numerical computations of spa
We show that a profinite group with the same first-order theory as the direct product over all odd primes $p$ of the dihedral group of order $2p$, is necessarily isomorphic to this direct product.
A Schur ring (S-ring) over a group $G$ is called separable if every of its similaritities is induced by isomorphism. We establish a criterion for an S-ring to be separable in the case when the group $G$ is cyclic. Using this criterion, we prove that
The aim of this short note is to provide a proof of the decidability of the generalized membership problem for relatively quasi-convex subgroups of finitely presented relatively hyperbolic groups, under some reasonably mild conditions on the peripher
We prove that any Cayley graph $G$ with degree $d$ polynomial growth does not satisfy ${f(n)}$-containment for any $f=o(n^{d-2})$. This settles the asymptotic behaviour of the firefighter problem on such graphs as it was known that $Cn^{d-2}$ firefig