International Journal of Group Theory
Volume:11 Issue: 3, Sep 2022

We exhibit presentations of the Von Dyck groups $D(2, 3, m), \ m\ge 3$, in terms of two generators of order $m$ satisfying three relations, one of which is Artin's braid relation. By dropping the relation which fixes the order of the generators we obtain the universal covering groups of the corresponding Von Dyck groups. In the cases $m=3, 4, 5$, these are respectively the double covers of the finite rotational tetrahedral, octahedral and icosahedral groups. When $m\ge 6$ we obtain infinite covers of the corresponding infinite Von Dyck groups. The interesting cases arise for $m\ge 7$ when these groups act as discrete groups of isometries of the hyperbolic plane. Imposing a suitable third relation we obtain three-relator presentations of $\text{PSL}(2,m)$. We discover two general formulas presenting these as factors of $D(2, 3, m)$. The first one works for any odd $m$ and is essentially equivalent to the shortest known presentation of Sunday [J. Sunday, Presentations of the groups ${\rm SL}(2,\,m)$ and ${\rm PSL}(2,\,m)$, Canadian J. Math., 24 (1972) 1129--1131]. The second applies to the cases $m\equiv\pm 2\ (\text{mod}\ 3)$, $m\equiv \hskip -9pt/ \ 11(\text{mod}\ 30)$, and is substantively shorter. Additionally, by random search, we find many efficient presentations of finite simple Chevalley groups PSL($2,q$) as factors of $D(2, 3, m)$ where $m$ divides the order of the group. The only other simple group that we found in this way is the sporadic Janko group $J_2$.

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‎For a finite group $G$‎, ‎three of the positive integers governing its‎ ‎representation theory over $\mathbb{C}$ and over $\mathbb{Q}$ are‎ ‎$p(G),q(G),c(G)$‎. ‎Here‎, ‎$p(G)$ denotes the {\it minimal degree} of a‎ ‎faithful permutation representation of $G$‎. ‎Also‎, ‎$c(G)$ and $q(G)$‎ ‎are‎, ‎respectively‎, ‎the minimal degrees of a faithful representation‎ ‎of $G$ by quasi-permutation matrices over the fields $\mathbb{C}$‎ ‎and $\mathbb{Q}$‎. ‎We have $c(G)\leq q(G)\leq p(G)$ and‎, ‎in general‎, ‎either inequality may be strict‎. ‎In this paper‎, ‎we study the‎ ‎representation theory of the group $G =$ Hol$(C_{p^{n}})$‎, ‎which is‎ ‎the holomorph of a cyclic group of order $p^n$‎, ‎$p$ a prime‎. ‎This group is metacyclic when $p$ is odd and metabelian but not‎ ‎metacyclic when $p=2$ and $n \geq 3$‎. ‎We explicitly describe the set‎ ‎of all isomorphism types of irreducible representations of $G$‎ ‎over the field of complex numbers $\mathbb{C}$ as well as the‎ ‎isomorphism types over the field of rational numbers $\mathbb{Q}$‎. ‎We compute the Wedderburn decomposition of the rational group‎ ‎algebra of $G$‎. ‎Using the descriptions of the irreducible‎ ‎representations of $G$ over $\mathbb{C}$ and over $\mathbb{Q}$‎, ‎we‎ ‎show that $c(G) = q(G) = p(G) = p^n$ for any prime $p$‎. ‎The proofs‎ ‎are often different for the case of $p$ odd and $p=2$‎.

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In this article‎, ‎we prove that if a nontrivial symmetric $(v‎, ‎k‎, ‎\lambda)$ design admit a flag-transitive and point-primitive automorphism group $G$‎, ‎then the socle $X$ of $G$ cannot be a projective special unitary group of dimension five‎. ‎As a corollary‎, ‎we list all exist nineteen non-isomorphism such designs in which $\lambda\in\{1,2,3,4,6,12‎, ‎16‎, ‎18\}$ and $X=\text{PSU}_n(q)$ with $(n,q)\in\{(2,7),(2,9),(2,11),(3,3),(4,2)\}$‎.

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‎Heineken [‎H‎. ‎Heineken‎, ‎Nilpotent subgroups of finite soluble groups‎, Arch‎. ‎Math.(Basel)‎, ‎ 56 no‎. ‎5 (1991) 417--423‎.] studied the order of the nilpotent subgroups of the largest order of a solvable group‎. ‎We point out an error‎, ‎and thus refute the proof of the main result of [‎H‎. ‎Heineken‎, ‎Nilpotent subgroups of finite soluble groups‎, Arch‎. ‎Math.(Basel)}‎, 56 no‎. ‎5 (1991) 417--423‎].

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We prove that the following families of (infinite) groups have complemented subgroup lattice‎: ‎alternating groups‎, ‎finitary symmetric groups‎, ‎Suzuki groups over an infinite locally finite field of characteristic $2$‎, ‎Ree groups over an infinite locally finite field of characteristic~$3$‎. ‎We also show that if the Sylow primary subgroups of a locally finite simple group $G$ have complemented subgroup lattice‎, ‎then this is also the case for $G$‎.

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