Subgroups
Given a group G and a non empty subset of it S
if there exists an operation ◌ that is closed in S and if inverses exists that are also closed in S; S is said to be a subgroup.
Note: S is also a group.
Some Vocabulary:
The smallest subgroup of any group is just the set that contains only the neutral element. The largest is the whole group itself. These two groups are called trivial subgroups. All other subgroups are called proper subgroups.
If you have a finite group then the set of all possible combinations of a subset of its elements is called the subgroup generated by the elements. Example let G be a group and a,b,c and their inverses belong to G then any combinatoin such as
abc, bca, ba¹cb¹… form a subgroup S then S is the subgroup generated by a,b,c.
If you take a single element of a group then you can form a subgroup in the same manner by having combinations of itself and its inverse. example: abitrary element a of a group G
aaa,aaaaaaaaaaa,a¹a¹a,a¹a¹a¹,…
We call this subgroup generated by a cyclic subgroup of G. a is then called its genarator. we denote the cyclic subgroup generated by a as <a>
If instead we form a group in the manner above via a single element the group is called cyclic group
