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.
   • Reflexive: $a\equiv a$ since $aa^{-1}=e\in H$ (the identity element).
   • Symmetric: Suppose $a\equiv b$, so that $ab^{-1}\in H$. Then $(ab^{-1})^{-1}\in H$ (since $H$ is a subgroup), and $(ab^{-1})^{-1}=ba^{-1}$. So $b\equiv a$.
   • Transitive: Suppose $a\equiv b$ and $b\equiv c$. Then $ab^{-1}\in H$ and $bc^{-1}\in H$. By closure, $ab^{-1}bc^{-1}=ac^{-1}\in H$. So $a\equiv c$.
2. Describe the set of equivalence classes.

   The equivalence classes are the cosets of $H$. Recall that two cosets $aH$ and $bH$ are equal if and only if $ab^{-1}\in H$.

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. Yes and no. The question asks, in particular, if $aHbH=(ab)H$ is a well defined binary operation on $G/H$. The answer is: YES if $H$ is a normal subgroup of $G$ and not necessarily otherwise. For a counterexample, consider the cosets of your favorite non-normal subgroup of $S_3$ (hint: use $H=\{\epsilon, (1\ 2)\}$. We will prove in class tomorrow that if $H$ is normal, then $G/H$ is a group.