Banca de DEFESA: Gabriel Azevedo Miranda

Uma banca de DEFESA de MESTRADO foi cadastrada pelo programa.
STUDENT : Gabriel Azevedo Miranda
DATE: 21/07/2023
TIME: 10:00
LOCAL: PPGMAT
TITLE:

On the average order in finite groups


KEY WORDS:
Average order, sum of orders, soluble groups, simple groups.

PAGES: 91
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUMMARY:
noindent Let $o(G)$ be the average order of the elements of a finite group G defined as
$$
o(G)=\frac{\psi(G)}{|G|},
$$
where $\psi(G)$ is the sum of the orders of all elements of $G$. A conjecture proposed by A. Jaikin-Zapirain consists of: if $N$ is a normal subgroup of $G$, then $o(G) \ge o(N)^{1/2}$. That said, E. I. Khukhro, A. Moreto and M. Zarrin gave a negative answer to this conjecture. In this way, we aim to present the construction of the counterexamples accommodated by them. In addition, we will also discuss the implications of this conjecture especially a solubility
candidate that involves the concept of average order. The following says: If $o(G)<o(A_5)$, then G is solvable. This result has been proved by M. Herzog, P. Longobardi and M. Maj. Finally, we generalize the inequality $o(G) \ge o(Z(G))$, demonstrated by A. Jaikin-Zapirain, and reproduce the same idea for the inequality $\alpha(G) \le \alpha(Z (G))$, where $\alpha(G)$ is a function widely investigated by M. Garonzi and I. Lima. \\
\noindent {\bf Keywords}: Average order, sum of orders, soluble groups, simple groups.

COMMITTEE MEMBERS:
Interno - 1198222 - EMERSON FERREIRA DE MELO
Presidente - 1984613 - IGOR DOS SANTOS LIMA
Interno - 2255154 - MARTINO GARONZI
Externo à Instituição - MOHSEN AMIRI
Notícia cadastrada em: 07/07/2023 10:43
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