Return to Homework 5, Glossary, Theorems
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, $a \sim c$.