GATE | GATE-CS-2009 | Question 60

For the composition table of a cyclic group shown below

*

a

b

c



d

a

a

b

c

d

b

b

a

d

c

c

c

d

b

a

d

d

c

a

b

 

Which one of the following choices is correct?
(A) a, b are generators
(B) b, c are generators
(C) c, d are generators
(D) d, a are generators


Answer: (C)

Explanation:

Check for all:-
a1 = a ,
a2 = a * a = a
a3 = a2 * a = a * a = a
a is not the generator since we are not able to express 
other members of the group in powers of a

Check for c -
c1 = c
c2 = c * c = b
c3 = c2 * c = b * c = d
c4 = c2 * c2 = b * b = a
We are able to generate all the members of the group from c ,
Hence c is the generator

Similarly check for d


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