Group

A group is a non-empty set $G$ together with a binary operation on $G$, denoted as "$\cdot$", such that the following three requirements are satisfied

Associativity For all $a,b,c\in G$, one has $(a\cdot b)\cdot c=a\cdot (b\cdot c)$

Identity element There exists an identity element $e$ in $G$ such that, for every $a$ in $G$, one has $e\cdot a=a,a\cdot e=a$

Inverse element For each $a$ in $G$, there exists an element $b$ in $G$ such that $a\cdot b=e$ and $b\cdot a=e$, where $e$ is the identity element. $b$ is denoted as $a^{-1}$

Subgroup

Given a group $G$ under a binary operation $\ast$, a subset $H$ of $G$ is called a subgroup of $G$ if $H$ also forms a group under the operation $\ast$. More precisely, $H$ is a subgroup of $G$ if the restriction of $\ast$ to $H\times H$ is a group operation on $H$. This is often denoted $H\le G$, read as "$H$ is a subgroup of $G$".