Project ForgeDiscrete Mathematics
Set Theory & Abstract Algebra
GATE 2009
Hard
For the composition table of a cyclic group shown below (elements a,b,c,d with a as identity: a*a=a, a*b=b, a*c=c, a*d=d; b*b=a, b*c=d, b*d=c; c*c=b, c*d=a; d*d=b), which one of the following choices is correct?
- Aa, b are generators
- Bb, c are generators
- c, d are generators
- Dd, a are generators
Explanation
a is the identity (order 1). b*b=a means b has order 2 (not a generator of an order-4 group). c*c=b and b*b=a give c⁴=a with c²=b≠a, so c has order 4 — a generator. Similarly d*d=b gives d⁴=a with d²=b≠a, so d also has order 4 — a generator. So c and d are the generators.
Source: GATE CS 2009 (2009)
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