Homomorphism & Isomorphism of Group - GeeksforGeeks

26 May, 2021
Introduction :We can say that  “o” is the binary operation on set G if : G is an non-empty set & G * G = { (a,b) : a , b∈ G } and  o : G * G –> G. Here, aob denotes the image of ordered pair (a,b) under the function / operation o.Example – “+” is called a binary operation on G (any non-empty set ) if & only if : a+b ∈G ; ∀ a,b ∈G and a+b give the same result every time when added.Real example – ‘+’ is a binary operation on the set of natural numbers ‘N’ because a+b ∈ N ; ∀ a,b ∈N and a+b  a+b give the same result every time when added. 
Laws of Binary Operation :In a binary operation o, such that : o : G * G –> G on the set G is :1. Commutative –
 aob = boa ; ∀ a,b ∈G
Example :  ‘+’ is a binary operation on the set of natural numbers ‘N’. Taking any 2 random natural numbers , say 6 & 70, so here a = 6 & b = 70, a+b = 6 + 70 = 76 = 70 + 6 = b + aThis is true for all the numbers that come under the natural number.
2. Associative –
ao(boc) = (aob)oc ; ∀ a,b,c ∈G
Example :  ‘+’ is a binary operation on the set of natural numbers ‘N’. Taking any 3 random natural numbers , say 2 , 3 & 7, so here a = 2 & b = 3 and c = 7,LHS : a+(b+c) = 2 +( 3 +7)  = 2 + 10 = 12RHS : (a+b)+c = (2 + 3) + 7 = 5 + 7 = 12This is true for all the numbers that come under the natural number.
3. Left Distributive – 
ao(b*c) = (aob) * (aoc) ; ∀ a,b,c ∈G
4. Right Distributive –
 (b*c) oa = (boa) * (coa)  ; ∀ a,b,c ∈G
5. Left Cancellation –
 aob =aoc  => b = c  ; ∀ a,b,c ∈G
6. Right Cancellation –
 boa = coa  => b = c ; ∀ a,b,c ∈G
Algebraic Structure :A non-empty set G equipped with 1/more binary operations is called an algebraic structure. Example : a. (N,+)  and b. (R, + , .), where N is a set of natural numbers & R is a set of real numbers. Here ‘ . ‘ (dot) specifies a multiplication operation. 
GROUP : An algebraic structure (G , o) where G is a non-empty set & ‘o’ is a binary operation defined on G is called a Group if the binary operation “o” satisfies the following properties –
1. Closure – 
a ∈ G ,b ∈ G  => aob ∈ G ;  ∀ a,b ∈ G
2. Associativity –
 (aob)oc = ao(boc) ; ∀ a,b,c ∈ G.
3. Identity Element – There exists e in G such that aoe = eoa = a ; ∀ a ∈ G (Example – For addition, identity is 0)
4. Existence of Inverse – For each element a ∈ G ; there exists an inverse(a-1)such that :  ∈ G such that – aoa-1 = a-1oa = e
Homomorphism of groups :Let (G,o) & (G’,o’) be 2 groups, a mapping “f ” from a group (G,o) to a group (G’,o’) is said to be a homomorphism if –
f(aob) = f(a) o' f(b) ∀ a,b ∈ G
The essential point here is : The mapping f : G –> G’ may neither be a one-one nor onto mapping, i.e, ‘f’ needs not to be bijective.
Example –If (R,+) is a group of all real numbers under the operation ‘+’ & (R -{0},*) is another group of non-zero real numbers under the operation ‘*’ (Multiplication) &  f is a mapping from (R,+) to (R -{0},*),  defined as : f(a) = 2a ;  ∀ a ∈ RThen f is a homomorphism like –  f(a+b) = 2a+b = 2a * 2b = f(a).f(b) . So the rule of homomorphism is satisfied & hence f is a homomorphism.
Homomorphism Into – A mapping ‘f’, that is homomorphism & also Into.
Homomorphism Onto – A mapping ‘f’, that is homomorphism & also onto.
Isomorphism of Group :Let (G,o) & (G’,o’) be 2 groups, a mapping “f ” from a group (G,o) to a group (G’,o’) is said to be an isomorphism if –
1. f(aob) = f(a) o' f(b) ∀ a,b ∈ G
2. f is a one- one mapping
3. f is an onto mapping.
If ‘f’ is an isomorphic mapping, (G,o) will be isomorphic to the group (G’,o’) & we write :
G ≅ G'
Note : A mapping f: X -> Y is called :
One – One  – If x1 ≠x2, then f(x1) ≠ f(x2) or if f(x1) = f(x2) => x1 = x2. Where x1,x2 ∈ XOnto           – If every element in the set Y is the f-image of at least one element of set X. Bijective      – If it is one one & Onto.
One – One  – If x1 ≠x2, then f(x1) ≠ f(x2) or if f(x1) = f(x2) => x1 = x2. Where x1,x2 ∈ X
Onto           – If every element in the set Y is the f-image of at least one element of set X. 
Bijective      – If it is one one & Onto.
Example of Isomorphism Group –If G is the multiplicative group of 3 cube-root units , i.e., (G,o) = ( {1, w, w2 } , *) where w3 = 1 &  G’ is an additive group of integers modulo 3 –  (G’, o’) = ( {1,2,3) , +3). Then : G ≅ G’ , we say G is isomorphic to G’.
The structure & order of both  the tables are same. The mapping ‘f’ is defined as :f : G -> G’ in such a way that f(1) = 0 , f(w) = 1 & f(w2) = 2.
Homomorphism property : f(aob) = f(a) o’ f(b) ∀ a,b ∈ G . Let us take a = w & b = 1LHS : f(a * b) = f( w * 1 ) = f(w) = 1.RHS : f(a) +3 f(b) = f(w) +3 f(1) = 1 + 0 = 1=>LHS = RHS
This mapping f is one-one & onto also, therefore, a homomorphism.
Engineering Mathematics
GATE CS
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.
Activation Functions
Difference between Propositional Logic and Predicate Logic
Logic Notations in LaTeX
Univariate, Bivariate and Multivariate data and its analysis
Z-test
Layers of OSI Model
ACID Properties in DBMS
TCP/IP Model
Types of Operating Systems
Normal Forms in DBMS