Let $(G, \cdot)$ be a group with an Identity element $e$.
(i) A relation on $G$ is defined through $g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$. Show that $\sim$ is a equivalence relation and show, that the equivalence class has for $[g]: ~[g]=\{g,g^{-1}\}$
(ii) Let now be $\#G$ even. Show that then $g \in G \setminus\{e\}$ exists with $g^2=e$
I have several problems solving this exercise, one is to write down this correctly To (i): I know i have to show Reflexivity, Symmetry and Transitivity.
Reflexivity: $g\sim g $ is true because $g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\} \Longrightarrow g\sim g$
How to go on with Symmetry and Transitivity?
For symmetry again go by the definition. Take any two elements $h,\,g\in G$ such that $h\tilde{} g$. Then $g\in \{h,\,h^{-1}\}$. Now there are two cases.
Case I: $g=h$. Then it's obvious that, $g\tilde{}h$
Case II: $g=h^{-1}$. Then $h=g^{-1}\Rightarrow h\in\{g,\,g^{-1}\}\Rightarrow g\tilde{} h$.
Hope you can continue with the the transitivity.