Q1: Let $(G, \circ)$ be a group. Is $\varphi: G \rightarrow G$ given by $\varphi(x) = x$ a group homomorphism?

A1: Yes, it always is.

A2: No, there are counterexamples.

A3: No, never.

Q2: Let $(G, \circ)$ and $(H, \ast)$ be two groups. What is not always correct for a group homomorphism $\varphi: G \rightarrow H$?

A1: $\varphi$ is injective

A2: $\varphi(x \circ y) = \varphi(x) \ast \varphi(y)$

A3: $\varphi(e_G) = e_H$

A4: $\varphi(a)^{-1} = \varphi(a^{-1})$

A5: $\varphi(a^{-1}) \ast \varphi(a) = e_H $

Q3: Let $({ a,b }, \circ)$ and $({ c, d }, \ast)$ be two groups. What is always correct for a group homomorphism $\varphi: G \rightarrow H$?

A1: $\varphi$ is injective.

A2: If $\varphi$ is injective, then $\varphi$ is bijective.

A3: $\varphi$ is surjective.

A4: $\varphi$ is bijective.

A5: If $\varphi$ satisfies $\varphi(a^{-1}) = \varphi(a)^{-1}$, then $\varphi$ is bijective.