Page 39 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes

Is this operation well defined? To check this, we have to see that if a  b  c d in Z , then
n

a b  c d.

Now, a  b (mod n) and c  d (mod n). Hence, there exist integers k and k such that a – b = k n
2
1
l
and c – d = k n. But then (a + c) – (b + d) = (a – b) + (c – d) = (k + k )n.
2
2
1
c

 a   b d .
Thus, + is a binary operation on Z.
For example, 2  2  0 in Z since 2 + 2 = 4 and 4  0(mod 4).
4
Now, let us show that (Z , +) is a commutative group.
n
(i) a  b   b  b a  b a  a, b Zn, i.e.,
a


addition is commutative in Z
(ii) a   b   c    b   c  a (b  c)

 a b  c (a b) c (a b) c        a, b, c Z ,

n
i.e., addition is associative in Z,.
(iii) a + 0 = a = 0 + a a Z,, i.e., 0 is the identity for addition,


(iv) ~ or ;  Z ,  n  Z such that a + n – a = n = 0 = n – a+ a.
n
n
Thus, every element a in Z has an inverse with respect to addition.
n
The properties (i) to (iv) show that (Z , +) is an abelian group.
n
Actually we can also define multiplication on Z by a . b = ab. Then, b = b a  a, b Z . Also,
n
n
(a b)c  a(b c)  a, b, c Z . Thus, multiplication in Z is a commutative and associative binary
n
n
operation.
Z, also has a multiplicative identity, namely, 1.

But (Z,, .) is not a group. This is because every element of Z , for example Q, does not have a
n
multiplicative inverse.


But, suppose we consider the non-zero elements of Z , that is, Z ,. . Is this a group? For example,
*
n n

*
*
Z   1, 2, 3 is not a group because is not even a binary operation on Z , since
*
4
4

*
*
2 . 2  0 Z . But  Z ,. is an abelian group for any prime p.

4

2.4.2 The Symmetric Group
We will now discuss the symmetric group briefly. In Next Unit we will discuss this group in
more detail.
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