Subgroup: the subsets in group theory.


In our last lesson, we gave a brief analogy of groups, in this lesson we will be discussing subgroups, how they are formed and their properties. 
If you are conversant with set theory and subsets, then to study subgroups will be an easy ride because i always say, to study groups, all you have to do is transfer whatever knowledge you have acquired on the study of sets to groups and you will do just fine, although there are exceptions to this relationship.

What is a subgroup? 
If $G$ is a group under a binary operation $"*" $, a subset $H$ of $G$ is called a subgroup if $H$ itself forms another group under the same operation $"*"$. This is denoted by $H\leq{G}$ which translates to $H$ is a subgroup of $G$. 
There are particular subgroups that form under a rather strict condition, although, this conditions does not except there ability to be called subgroups, example, in the case of the trivial subgroups of a group, these are subgroups that contains only the identity element $e$. 
Remember in set theory, the proper subset is a set that is not equal to its superset, we also have a proper subgroup in group theory, a proper subgroup is a subgroup that is not equal to its group, if $G$ is a group and $H$ a subgroup of $G$, then $H$ is a proper subgroup if $H$ is not equal to $G$. This is denoted as $H

Some basic properties of subgroup 
  1.  The group $G$ and its subgroup $H$ all shares the same identity element $e$, i.e $e_G=e_H$. 
  2.  The group $G$ and its subgroup $H$ also shares the same inverse element. 
  3. For any two subgroups $A$ and $B$, the intersection of $A$ and $B$ is also a subgroup. 
  4.  For any two subgroups $A$ and $B$, the union of $A$ and $B$ is also a subgroup if and only if either $A$ is contained in $B$ or $B$ is contained in $A$. 
  5. Every element a of a group $G$ generates the cyclic subgroup $⟨a⟩$.