📂Abstract Algebra

Definition and Test Method of Subgroups

Definition ¹

A subset $H$ of a group $G$ is called a subgroup of $G$ if $H$ itself is a group under the operation of $G$.

Theorem

Subgroup Test: For a non-empty subset $H$ of a group $G$, if for every element $a,\ b$ in $H$, $ab^{-1}$ is also an element of $H$, then $H$ is a subgroup of $G$. In other words, if whenever $a,\ b$ is in $H$, $a-b$ is also in $H$, then $H$ is a subgroup.

Proof

Assume that whenever $a,\ b$ is an element of $H$, $ab^{-1}$ is also an element of $H$. We need to verify whether $H$ satisfies the three conditions to be a group.

1. The operation of $H$ being the same as that of $G$, associativity is trivially satisfied.
2. Let’s assume $a=x,\ b=x$. Then $ab^{-1}=xx^{-1}=e$ and by assumption, it’s an element of $H$, which means $H$ has an identity element.
3. Assume $a=e,\ b=x$. Then $ex^{-1}=x^{-1}$ and by assumption, it becomes an element of $H$, which means any element $b$ in $H$ has an inverse.
4. By 3, having verified that every element has an inverse, assume $a=x,\ b=-y$. Then $x(y^{-1})^{-1}=xy$ and by assumption, it becomes an element of $H$, which means $H$ is closed under the operation.

By points 1-4, since $H$ is closed under the operation of $G$, satisfies associativity, has an identity element, and every element has an inverse, it is a group. Therefore, the subset $H$ is a subgroup of the group $G$.

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