Introduction to group theory/Part 2 Subgroups and cyclic groups
Introduction
[edit | edit source]We will learn briefly about subgroups and cyclic groups. As per the name suggest "sub" means a part of i.e; heading-subheading ,set- subset etc etc.
So subgroup is a part of a group which forms under the same binary operation as per the group. [ Note: all elements of subgroup must be elements of groups]
Definition
[edit | edit source]A non-empty subset H of a group G is said to be subgroup of G if , under the product in G, H itself forms a group.
i.e;
It should follow all the properties of groups.
- Closure
- Associative
- Indentity
- Inverse
- Commutative (optional)
The following remark is clear: if H is a subgroup of G and K is a subgroup of H , then K is a subgroup of G.
LEMMA
[edit | edit source]A non empty subset H of the group G is a subgroup of G if and only if
Proof
[edit | edit source]If H is a subgroup of G , then it is obvious that (1) and (2) must hold.
In order to establish that H is subgroup, all that is needed is to verify that and that the associative law holds for elements of H .
Since associative law does hold for G, it holds all more so for H, which is a subset of G.
if , by part 2, and so by part 1 ,
LEMMA
[edit | edit source]A non-empty finite subset H of the group G is a subgroup of G if and only if
1.
Proof
[edit | edit source]The Proof is similar to the one for the previous lemma and left as an exercise for the reader.
Cyclic Group
[edit | edit source]A group is said to be a cyclic group , if there exist an element such that every element of G can be expressed as some power of a.
If G is a group generated by 'a' . we can say that a is a generator of G and all the elements of G can form by some power of a.
G = (a) { Here a is the generator}
Notes
[edit | edit source]- A cyclic group may have more that one generator.
- Every group has two trivial subgroups.
- The group containing all elements.
- The group containing identity element only.
Congruent Modulo of a Subgroup
[edit | edit source]Let H is a subgroup of G.( )
if
LEMMA
[edit | edit source]is an equivalence relation.
proof
[edit | edit source]An equivalence relation must follows 3 properties.
- Reflexive
- Symmetric
- Transitive
Reflexive
[edit | edit source]let
as
Symmetric
[edit | edit source]let such that
we have to show that
[ Subgroup follows closure law]
Transitive
[edit | edit source]let such that
and
and
[Closure]
[Associative]
[Identity]
modulo relation is an equivalence relation.