Um estudo sobre automorfismos potências com centralizadores 2-grupos abelianos elementares
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2022-09-09
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Universidade Federal de Goiás
Resumo
Let $\varphi$ be an automorphism of a group $G$. We denote by $C_G(\varphi)$ the centralizer of $\varphi$ in $G$, that is, the subgroup of the fixed points of $\varphi$ to $G$. It is known that various properties of $G$ are in a certain sense close to the corresponding properties of the subgroup $C_{G}(\varphi)$. In the case where $\varphi$ is a power automorphism, we have that all elements having order 2 are fixed by $\varphi$. For this reason, we consider the case where $C_{G}(\varphi)$ is an elementary abelian $2$-group. A power automorphism $\varphi$ is said to be a pre-fixed-point-free power automorphism if $C_{G}(\varphi)$ is an elementary abelian $2$-group. When a group $G$ admits a pre-fixed-point-free power automorphism, we say that $G$ is an $E$-group. In this work, we determine all $E$-groups and their pre-fixed-point-free by power automorphisms. In particular, we use some results on power automorphisms to show a characterization of finite abelian groups.
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SOUZA, Gabriella Cristina. Um estudo sobre automorfismos potências com centralizadores 2-grupos abelianos elementares. 2022. 96 f. Dissertação (Mestrado em Matemática) - Universidade Federal de Goiás, Goiânia, 2022.