Property of defect diminishing and stability

Authors : Marco Antonio Garcia MORALES, Lev GLEBSKY

Pages : 49-54

Doi:10.24330/ieja.1058399

View : 15 | Download : 8

Publication Date : 2022-01-17

Article Type : Research Paper

Abstract :Let $\Gamma$ be a group and $\mathscr{C}$ a class of groups endowed with bi-invariant metrics. We say that $\Gamma$ is $\mathscr{C}$-stable if every $\varepsilon$-homomorphism $\Gamma \rightarrow G$, $insert ignore into journalissuearticles values(G,d); \in \mathscr{C}$, is $\delta_\varepsilon$-close to a homomorphism, $\delta_\varepsilon\to 0$ when $\varepsilon\to 0$. If $\delta_\varepsilon < C \varepsilon$ for some $C$ we say that $\Gamma$ is $ \mathscr{C} $-stable with a linear rate. We say that $\Gamma$ has the property of defect diminishing if any asymptotic homomorphism can be changed a little to make errors essentially better. We show that the defect diminishing is equivalent to the stability with a linear rate.
Keywords : Group theoretic stability, ,