Find an element a in Z7 such that every nonzero element of Z

Find an element a in Z_7 such that every nonzero element of Z_5. Can you do part (a) in Z_6?

Solution

(a). Z7={0,1,2,3,4 ,5,6}

There are a=1,2,3,4,5,6 all non zero elements in Z7

Such that every non zero elements are power of a \'s

1=1^1=1

2=1^2=2*1=2

.

.

6=1^6=6*1=6

1=2^4=4*2=8=1

2=2^1=2

3=2^5=10=3

4=2^9=9*2=18=11=4

5=2^6=6*2=12=5

6=2^10=20=13=6

Similarly for a=3

1=3

2=3^3=9=2

3=3^1=3

4=3^6=18=4

5=3^4=12=5

6=3^9=27=6

Similarly we can write for all a\'s.

(b). Z5={0,1,2,3,4}

Since 5 is prime number there are a\'s=1,2,3,4 all non identity elements of Z7 such that all elements of Z7 can be written as power of a\'s .

And these elements are called generator for the group .

(c). For Z6={0,1,2,3,4,5}

There are only two a\'s 1 and 5 which are relatively prime to 6 such that every nonzero element s of Z6 can be written as power of a =1 and 5 .

For 1 is trivial

For a=5

1=5^5=25=1

2=5^4=20=2

3=5^3=15=3

4=5^2=10=4

5=5^1=5