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

GATE 2009 · Discrete Mathematics · Group Theory · medium

Answer: c and d are the generators, so option C is correct.

1. Find the identity: Row a is a b c d, identical to the column headers, so a is the identity element e.
2. Order of b: b*b = a = e, so order(b) = 2. An order-2 element cannot generate a 4-element group.
3. Order of c and d: c^2 = c*c = b, c^3 = c*b = d, c^4 = c*d = a = e, so order(c) = 4 = |G|. Similarly d*d = b, d^3 = d*b = c, d^4 = a, so order(d) = 4.