Page 38 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes if {a} = x : x I and x – a is divisible by p}.
If p | a then {a} = {0} which is called the zero residue class. Let G be the set of non-zero residue
classes mod p (p being prime) then

G = {1, 2, 3, … (p – 1)}
Closure axiom: Let r , r G then r r r (mod p)
1 2 1 2
where r is the least non-negative integer such that 0 < r < p – 1 obtained after dividing r , r by p.
1 2
Also, since p is prime, r , r is not divisible by p. Hence r cannot be zero.
1 2
Hence, r r 2 r G .
1
Thus closure axiom is satisfied.
Associative axiom: Multiplication of residue classes is associative.

Existence of Identity: 1 G and a. 1 = 1 a = a a G.
Therefore 1 is the identity element in G with respect to multiplication.

Existence of Inverse: Let s G then 1 s p 1 . Let us consider following (p – 1) elements.
s
s
1 , 2 , 3 , ,(p 1) s .
s
All these elements are elements of G because the closure law is true. All these elements are
distinct as otherwise if
i
i s j s for i j and , j G
the i s j s i s j s is divisible by p

(i ) j s is divisible by p

(i – j) is divisible by p [because 1 s (p 1) ]
i – j which is contrary to our assumption that i j.

Therefore above (p – 1) elements are the same as the elements of G. Hence some one of them
should be 1 also, let s s 1 where 1 s p 1 . Hence s’ is inverse of s. Hence inverse axiom is
also satisfied.
G is a group under multiplication mod p.

Note: Since r . s = s . r r, s G.
G is finite abelian group of order (p – 1).

Illustrative Examples

Example 10: Prove that the set G = {0, 1, 2, 3, 4} is a finite abelian group of order 5 with
respect to addition modulo 5.

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