Group Theory
Binary operation on sets:
A binary operation on a set S is a mapping (function) from SxS to S. As function, a binary operation should map an element in SXS to one and only one element in S. Generally, these binary operations will be denoted by ‘*’.* :SXS→S
Equivalently, for any (a, b)∈SXS there exist an element c∈S such that, a*b = c.Example:
The usual multiplication ‘x’ on the set of natural numbers N= {1, 2, 3, …} is a binary operation. Symbolically,x :NXN→N
and for any pair of numbers we may find their multiplication which is a number in N.Definition of Group:
A set G together with a binary operation ‘*’ defined on it is said to be a group if it satisfies the following axioms.Closure property:
There must exist an element x*y ∈ G, for every x, y ∈ G
Associative property:
x*(y*z) = (x*y)*z, for every x, y, z ∈G
Identity property:
There exist an element e∈G such that,e*x = x*e = x, for every x∈G
Inverse property:
There exist an element x-1∈G for every x∈G such that,x*x-1 = x-1*x = e
Then the group G under * is denoted by the pair (G, *).Abelian Group or Commutative Group:
A group (G, *) which in addition satisfies the commutative law,x*y = y*x, for every x, y∈G
is said to be an abelian group or a commutative group.More on Groups:
A set G together with a binary operation ‘*’ defined on it is said to be a semi-group if it satisfies only Closure and Associative properties of Groups.
A set G together with a binary operation ‘*’ defined on it is said to be a monoid if it satisfies Closure, Associative and Identity properties of Groups.
<a> = {a, a*a, a*a*a, …} = {an, for any integer n} (multiplicative group) <a> ={a, a+a, a+a+a, …}={na, for any integer n} (additive group)
is said to be the Cyclic group generated by ‘a’ and ‘a’ is said to be the generator of <a>.Example:
Consider the set A = {i, -i, 1, -1}. Take a =i then i2 = -1, i3 = -i, i4 = 1. Then we may write, <i>={i, -i, 1, -1}. Easily we may prove that (A,*) is a group. This group is said to be the cyclic group generated by i whose order is 4 since i4 = 1, the multiplicative identity and ‘i’ is said to be the generator of <i>.i.e., G = <a>, for some a∈G
n is said to be the order of G if an=e, the identity element.A subset H of a group (G,*) is said to be a subgroup of G if (H, *) itself forms a group. Equivalently, if H ⊂ G, and H satisfies Closure, Associative, Identity and Inverse properties under the same binary operation ‘*’, under which G is group then (H, *) is said to be a subgroup of (G,*).
Left Coset:
Let (H,*) be a subgroup of a group (G,*). Then for some x∈G, the set given by,xH={xh|h∈H}
is said to be the left coset of H in G.Right Coset:
Let (H,*) be a subgroup of a group (G,*). Then for some x∈G, the set given by,Hx={hx|h∈H}
is said to be the right coset of H in G.A subgroup (N,*) of the group (G,*) is said to be a normal subgroup of G iff
gng-1∈N, ∀g∈G and ∀n∈N
Homomorphisms of Groups:
Let (G,*) and (G1, ⊛) be groups. A mapping f between G and G1 is said to be a homomorphism if f preserves operations on groups. That is,
if f:G→G1 satisfies f(x*y) = f(x)⊛f(y), ∀x,y∈G
then f is said to be a homomorphism between G and G1. Equivalently, a mapping which maps ‘multiplication of elements in G’ to ‘the multiplication of their images in G1‘ is said to be a homomorphism between G and G1.
Isomorphism of Groups:
Let (G,*) and (G1, ⊛) be groups. A mapping f between G and G1 is said to be an isomorphism if,
- f is 1-1
- f is onto and
- f is a homomorphism.
Two groups (G,*) and (G1, ⊛) are said to be isomorphic if there exist an isomorphism between G and G1 .
Automorphism of Groups:
An isomorphism of a group (G,*) into itself is said to be an automorphism.