Worksheet 4

Please work on the following problems.

1. Let $G$ be a group and $H$ a subgroup of $G$. Define a relation on $a,b\in G$ by $a\equiv b$ iff $ab^{-1}\in H$. Is it an equivalence relation? Prove it.
2. Describe the set of equivalence classes.

3. Let $G/H$ denote the set of equivalence classes. Is $G/H$ a group using the binary operation from $G$? If yes, prove it. If no, provide a counterexample.