Page 38 - DMTH403_ABSTRACT_ALGEBRA

Unit 2: Groups

= [(a ) } , by the case n > 0 Notes
-a -1 -1
= ad

Also, (a ) = (a )
-1 n
-l -(-n)
= [(a ) ] , by the case n > 0
-1 -1 -n
= a*.
So, in this case too,

(a ) = a = (a ) .
-n
n -1
-1 n
(b) If m = 0 or n = 0, then a m+n = a . a . Suppose m  0 and n  0.
n
m
Case 1 (m > 0 and n > 0): We prove the proposition by induction on n.
If n = 1, then a . a = a m+1 , by definition.
m
Now assume that a . a = a m+n-1
n-1
m
Then, a . a = a (a . a) = (a . a ) a = a m+n-1 . a = a m+n . Thus, by the principle of induction, (a) holds
n-1
m
m
m
n
n-1
for all m > 0 and n > 0.
Case 2 (m < 0 and n < 0): Then (-m) 0 and (-n) > 0. Thus, by Case 1, a . a = a -(n+m) = ad ). Taking
-m
-n
wn
inverses of both the sides and using (a), we get,
g m+n = (a . a ) = (a ) . (a ) = a . a .
m
n
-n -1
-m -1
-m -1
-n
Case 3 (m > 0, n < 0 such that m + n  0): Then, by Case 1, a m+n . a = a . Multiplying both sides on
-n
m
the right by a = (a ) , we get a m+n = a . a .
-n -1
m
n
n
Case 4 ( m > 0, n < 0 such that m+n < 0): By Case 2, a . a m+n = an. Multiplying both sides on the left
-m
by a = (a ) , we get a m+n = a . a .
n
m
-m -1
m
The cases when m < 0 and n > 0 are similar to Cases 3 and 4. Hence, a = a . a for all a  G and
wn
m
n
m, n  Z.
2.4 Different Types of Group
2.4.1 Integers Modulo n
Consider the set of integers, Z, and n  N. Let us define the relation of congruence on Z by : a is
congruent to b modulo n if n divides a-b. We write this as a  b (mod n). For example, 4  1
(mod 3), since 3 | (4-1).
Similarly, (-5)  2(mod 7) and 30  0 (mod 6).
 is an equivalence relation, and hence partitions Z into disjoint equivalence classes called
congruence classes modulo n. We denote the class containing r by r.
Thus, r ={ m  Z | m  r (modn) }.
So an integer m belongs to r for some r, 0  r < n, iff n | (r-m), i.e., iff r–m = kn, for some k  Z.
 r = { r + kn | k 
Now, if m  n, then the division algorithm says that m = nq + r for some q, r  Z, 0  r < n. That
is, m  r (mod n), for some r = 0, .,..., n-1. Therefore, all the congruence classes modulo n are
0,1, ....., n 1. Let Z  {0,1,2, ....., n  1}. We define the operation + on Z by a b a b.  

n
n
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