Recent zbMATH articles in MSC 20C20https://www.zbmath.org/atom/cc/20C202021-04-16T16:22:00+00:00WerkzeugThe homogeneous pseudo-embeddings and hyperovals of the generalized quadrangle \(H(3,4)\).https://www.zbmath.org/1456.510032021-04-16T16:22:00+00:00"De Bruyn, Bart"https://www.zbmath.org/authors/?q=ai:de-bruyn.bart"Gao, Mou"https://www.zbmath.org/authors/?q=ai:gao.mouLet \(\mathcal{S}=(\mathcal{P},\mathcal{L})\) be a generalized quadrangle and denote by \(\varepsilon:\mathcal{S} \longrightarrow \mathrm{PG}(V)\) the universal pseudo-embedding \(\mathcal{S}.\) The vector dimension of the subspace \(\langle \varepsilon(\mathcal{P})\rangle\) of \(V\) is called the \textit{pseudo-embedding rank} of \(\mathcal{S}.\)
If \(|\mathcal{P}|\) is finite, denote by \(C=C(\mathcal{S})\) the binary code of length \(|\mathcal{P}|\) generated by the characteristic vectors of the lines of \(\mathcal{S}.\) It has been proved in [the first author, Adv. Geom. 13, No. 1, 71--95 (2013; Zbl 1267.51002)] that the pseudo-embedding rank of \(\mathcal{S}\) is \(|\mathcal{P}|- \dim(C).\)
In this paper, the authors prove that the pseudo-embedding rank of the Hermitian quadrangle \(H(3,4)\) is equal to \(24.\) As a consequence, the binary code \(C(H(3,4))\) has dimension \(45-24=21,\) because the generalized quadrangle \(H(3,4)\) has \(45\) points.
They also show that there are, up to isomorphism, four homogeneous pseudo-embeddings of \(H(3,4),\) with respective vector dimensions \(14,\) \(15,\) \(23\) and \(24.\)
Reviewer: Guglielmo Lunardon (Napoli)Large orbit sizes in finite group actions.https://www.zbmath.org/1456.200062021-04-16T16:22:00+00:00"Qian, Guohua"https://www.zbmath.org/authors/?q=ai:qian.guohua"Yang, Yong"https://www.zbmath.org/authors/?q=ai:yang.yongLet \(G\) be a finite group acting faithfully on a finite vector space \(V\). The \(G\)-orbit of an element \(v \in V\) is the set \(v^{G}=\{v^{g} \mid g \in G \}\) and the orbit size of \(v\) is \(| v^{G}|\). Results on orbit sizes, particularly on the existence of large orbits, have been fundamental to solving problems in several areas of finite group theory.
In the paper under review, the authors study relations of the sizes of various sections of finite linear groups and the largest orbit size of linear group actions. They also provide various applications of the results they have obtained.
Reviewer: Enrico Jabara (Venezia)Algebras of generalized dihedral type.https://www.zbmath.org/1456.160082021-04-16T16:22:00+00:00"Erdmann, Karin"https://www.zbmath.org/authors/?q=ai:erdmann.karin"Skowroński, Andrzej"https://www.zbmath.org/authors/?q=ai:skowronski.andrzejSummary: We provide a complete classification of all algebras of generalized dihedral type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with dihedral defect groups. This gives a description by quivers and relations coming from surface triangulations.Landau's theorem for \(\pi\)-blocks of \(\pi\)-separable groups.https://www.zbmath.org/1456.200072021-04-16T16:22:00+00:00"Sambale, Benjamin"https://www.zbmath.org/authors/?q=ai:sambale.benjaminLet \(\pi\) be a set of primes. The main result of the paper shows that the order of a defect group of a \(\pi\)-block \(B\) of a \(\pi\)-separable group can be bounded by a function depending only on \(k(B)\), the number of irreducible characters in \(B\). This generalizes a result by the reviewer [J. Reine Angew. Math. 404, 189--191 (1990; Zbl 0684.20007)] on \(p\)-blocks of \(p\)-solvable groups where \(p\) is a single prime. The author also works out explicitly the cases where \(k(B) \le 3\).
Reviewer: Burkhard Külshammer (Jena)