Hw5 Problem 9

### Problem 9

(a) Show that a nonempty subset $H$ of a group $G$ is a subgroup of $G$ if and only if $ab^{-1} \in H$ for all $a,b \in H$.
(b) Let $H$ be a subgroup of a group $G$. For $a,b \in G$, define $a \sim b$ if and only if $ab^{-1} \in H$. Show that $\sim$ defines an equivalence relation.

Solution

(a)
We need to show that
if $H \leq G$ with $a,b \in H$, then $ab^{-1} \in H$
and that
if for all $a,b \in H, ab^{-1} \in H$ as well, then $H \leq H$.

Suppose that $H \leq G$ with $a,b \in H$.
Then $b^{-1} \in H$, by definition of subgroup.
Hence, $ab^{-1} \in H$, since subgroups are closed under the operation.

Now, let $a,b \in H$ and suppose $ab^{-1} \in H$.

(b)
Reflexive
Suppose $a \in H$. Now, $aa^{-1}$ is in $H$ because $aa^{-1}=e$ and since $H$ is a subgroup, $H$ must contain the identity. Hence, $a \sim a$.
Symmetric
Suppose $a \sim b]]. Then [[$ab^{-1} \in H$. Since$H$is a subgroup,$(ab^{-1})^{-1} \in H$. That is,$a^{-1}b \in H$. Hence,$b \sim a$. Transitive Suppose$a\sim b$and$b\sim a$. Then$ab^{-1} \in H$and$bc^{-1} \in H$. Since$H$is closed by definition of subgroup$(ab^{-1})(bc^{-1}) \in H$. By associativity,$a(b^{-1}b)c^{-1} \in H$. That is,$ac^{-1} \in H$. Hence,$a \sim c\$.