Here is the translation of the text:

Problem 1 (10 points)
Let $(G, \circ)$ be a group.

a) Show that for all $x, y \in G$ the following holds:
$(x \circ y)^{-1} = y^{-1} \circ x^{-1}.$

b) Find a group $(G, \circ)$ and elements $x, y \in G$ such that
$(x \circ y)^{-1} \neq x^{-1} \circ y^{-1}.$

c) Let $f, f' \in G$ be elements that satisfy the equations
$f \circ g = g \quad \text{and} \quad g \circ f' = g$ for all $g \in G$. Show that $f = f'$ is the neutral element of the group $G$.

Would you like detailed solutions or explanations for any part? Here are 5 questions that expand on this topic:

1. What is the general property of inverses in a group?
2. How can you construct a group where $(x \circ y)^{-1} \neq x^{-1} \circ y^{-1}$?
3. What is the significance of the neutral element in group theory?
4. How do left and right inverses relate to each other in a group?
5. Can you provide other examples where $f \circ g = g$ leads to identifying the neutral element?

Tip: Always verify group axioms (closure, associativity, identity, and inverses) when working with unfamiliar groups.