MAT Notícias

Seminário de Álgebra -  sexta-feira, 17/05, às 14h30

O seminário ocorrerá no próximo dia 17/05, às 14h30, no auditório do MAT.

 
Abstract:
 
A finite group $G$ is called pyramidal if all involutions of $G$ are conjugated in $G$, and $G$ is called $m$-pyramidal if it is pyramidal and it has precisely $m$ involutions, where $m$ is a positive integer. Pyramidal groups can be interpreted as specific groups of automorphisms of certain combinatorial structures called pyramidal Kirkman triple systems. More specifically, an $m$-pyramidal group acts as automorphism group of an $m$-pyramidal Kirkman triple system, regularly on all but $m$ fixed points. Obviously, the order of an $m$-pyramidal group $G$ is strongly related to the vertex size $X$ of an $m$-pyramidal Kirkman triple system. We will prove that, if $m$ is an odd prime power $p^k$, then every $m$-pyramidal group is solvable if and only if either $m=9$ or $k$ is odd. We also determine the orders of the $m$-pyramidal groups when $m$ is a prime number. Moreover, we obtain a classification of $3$-pyramidal groups. Subsequently, the numbers of cyclic and maximal cyclic subgroups of $G$ are discussed. A family of groups is called (maximal) cyclic bounded ((M)CB) if, for every natural number $n$, there are only finitely many groups in the family with at most $n$ (maximal) cyclic subgroups. In this topic we will prove that the family of groups of prime power order is MCB. We also prove that the family of finite groups without cyclic coprime direct factors is CB. As a consequence, a natural number $n \geqslant 10$ is prime if and only if there are only finitely many finite groups with precisely $n$ cyclic subgroups.

Últimas Notícias

O seminário ocorrerá no próximo dia 17/05, às 14h30, no auditório do MAT.

O seminário ocorrerá no próximo dia 17/05, às 14h30, no auditório do MAT.

Vai acontecer no MAT o seminário de Jogos Matemáticos: Experiência Prática 

O IV Ciclo de Palestras: Residência Pedagógica do MAT/UnB ocorrerá de 06 a 10/05, sempre às 18h, no Auditório da Matemática/UnB e no Youtube.