Trivial Representation 418Ea2

1. **Problem Statement:** We consider the trivial representation of a group $G$, which maps every element of $G$ to the identity transformation. 2. **Definition of Trivial Representation:** The trivial representation $\rho : G \to \mathrm{GL}(V)$ is defined by $\rho(g) = I$ for all $g \in G$, where $I$ is the identity map on the vector space $V$. 3. **Definition of Faithful Representation:** A representation $\rho$ is faithful if it is injective, i.e., if $\rho(g) = I$ implies $g = 1_G$. 4. **Part (i): Show that the trivial representation is faithful if and only if $G = \{1_G\}$:** - Since $\rho(g) = I$ for all $g \in G$, the kernel of $\rho$ is all of $G$. - For $\rho$ to be faithful, the kernel must be $\{1_G\}$ only. - Therefore, $\ker(\rho) = G = \{1_G\}$. - Hence, the trivial representation is faithful if and only if $G$ contains only the identity element. 5. **Part (ii): Show that the trivial representation is not faithful if $|G| > 1$:** - If $|G| > 1$, then there exists $g \neq 1_G$ in $G$. - Since $\rho(g) = I$ for all $g$, the kernel contains more than just $1_G$. - Therefore, $\rho$ is not injective and hence not faithful. **Final answers:** - (i) The trivial representation is faithful if and only if $G = \{1_G\}$. - (ii) The trivial representation is not faithful if $|G| > 1$.