Homomorphism - Group Theory, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

EXAMPLE

If the operations on A and B are both addition, then the homomorphism condition is    If A and B are both rings, with addition and multiplication, there is also a multiplicative condition:  

DEFINATION

Let R and S be rings, with operations + and . (this is a slight abuse of notation, but the formulas below are more unwieldy with subscripts on the operations). A ring homomorphism f : R → S is a function f such that

(In this wiki, "ring" means "ring with unity"; a homomorphism of rings is defined in the same way, but without the third condition.)

EXAMPLE

Show that if f : R → S is a ring homomorphism, f(0_R) = os.

Note that    by the homomorphism property. Since f(0_R) has an additive inverse in S, we can add it to both sides of this equation to get 0_S =  f(0_R).

EXAMPLE

1. For any groups G and H, there is a trivial homomorphis 
2. Let  be a positive integer. The function is a ring homomorphism (and as such, it is a homomorphism of additive groups).
3. Define  is complex conjugation. Then c is a homomorphism from  C to itself. It is clearly a bijection, so it is in fact an isomorphism from C to itself.
4. Let R be a subring of S, and pick  Then there is an evaluation homomorphism  is the ring of polynomials with coefficients in R.
It is given by 
5. The map   is a group homomorphism. Note that R is an additive group and R* the set of nonzero real numbers, is a multiplicative group. The verification that f is a group homomorphism is precisely the law of exponents: 
6. Let S_n be the symmetric group on n letters. There is a unique nontrivial group homomorphism  the latter being a group under multiplication. The value  is called the sign of σ, and is important in many applications, including one definition of the determinant of a matrix.

DEFINATION

  be a group homomorphism. The kernel of f, ker (f), is the subset of G consisting of elements G such that  is the group identity element).

 be a ring homomorphism. The kernel of   is the subset of  R consisting of elements R such that 

EXAMPLE

Continuing the six examples above:

1. If   is the trivial homomorphism, then ker  the trivial subgroup of  H
2. The kernel of reduction mod n is the ideal  consisting of multiples of n. The image is all of Z_n; reduction mod n is surjective.
3. The kernel of complex conjugation is {0}, the trivial ideal of C (Note that 0 is always in the kernel of a ring homomorphism, by the above example.) The image is all of C.
4. The kernel of evaluation at α is the set of polynomials with coefficients in R which vanish at α. This ideal is not always easy to determine, depending on the nature of R and S. To take a common example, suppose   Which polynomials with rational coefficients vanish on  (See the algebraic number theory wiki for an answer.)
The image of evaluation at α is a ring called R[α], which is a subring of S consisting of polynomials in α with coefficients in R.
5. The kernel of exponentiation is the set of elements which map to the identity element of R*, which is 1 So the kernel is {0}. And the image of exponentiation is the subgroup  of positive real numbers.
6. The kernel of the sign homomorphism is known as the alternating group A_n. It is an important subgroup of S_n which furnishes examples of simple groups for   The image of the sign homomorphism is   since the sign is a nontrivial map, so it takes on both  for certain permutations.