Question

For the composition table of a cyclic group shown below:
$\begin{array}{ccccc}\ast & a& b& c& d\\ a& a& b& c& d\\ b& b& a& d& c\\ c& c& d& b& a\\ d& d& c& a& b\end{array}$
Which one of the following choices is correct?

• A. a, b are generators
• B. be c are generators
• C. c, d are generators
• D. d, a are generators

Answer 1

The correct option is C c, d are generators

If an element is a generator, all elements must be obtained as powers of that element.
Try $a,b,c,d$ one by one to see which are the generators.
$a=a$
$a^2=a.a=a$
$a^3=a.a^2=a.a=a$ and so on.
$\therefore a$ is not the generator.
$b=b$
$b^2=b.b=a$
$b^3=b.b^2=b.a=b$
$b^4=b.b^3=b.b=a$ and so on.
$\therefore b$ is not the generator
$c=c$
$c^2=c.c=b$
$c^3=c.c^2=c.b=d$
$c^4=c.c^3=c.d=a$
Since all of $a,b,c,d$ have been generated as powers of $c$
$\therefore c$ is a generator of this group
Similarly,
$d=d$
$d^2=d.d=b$
$d^3=d.d^2=d.b=c$
$d^4=d.d^3=d.c=a$
$\therefore d$ is the other generator.