GATE CSE 2009 | Question: 22

Question

For the composition table of a cyclic group shown below:
$$\begin{array}{|c|c|c|c|c|} \hline \textbf{*} & \textbf{a}& \textbf{b} &\textbf{c} & \textbf{d}\\\hline \textbf{a} & \text{a}& \text{b} & \text{c} & \text{d} \\\hline \textbf{b} & \text{b}& \text{a} & \text{d} &\text{c}\\\hline \textbf{c} & \text{c}& \text{d} & \text{b} & \text{a}\\\hline \textbf{d} & \text{d}& \text{c} & \text{a} & \text{b} \\\hline \end{array}$$
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

Comments

Good reads: http://sites.millersville.edu/bikenaga/abstract-algebra-1/cyclic-groups/cyclic-groups.pdf

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Without checking the Cayley table,we can answer it directly by inspecting the options. Here, we see a is an identity element,so it can't be generator,hence throw option A and D right now. Generators can't exist in isolation,so if g is generator,so is g inverse. We see C and D are inverses of each other ,so it has to right  

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$\color{Red}\text{Understand the solution:}$

https://youtu.be/83GuHnqT8UA?list=PLIPZ2_p3RNHhXves0XVa8d5O6F4rUi3KR&t=1786

Answer 1

Check for all:-
a^1 = a ,
a^2 = a * a = a
a^3 = a^2 * 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 -
c^1 = c
c^2 = c * c = b
c^3 = c^2 * c = b * c = d
c^4 = c^2 * c^2 = 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

d^1 = d
d^2 = d * d= b
d^3 = d^2 * d = b * d = c
d^4 = d^2 * d^2 = b * b = a

Option (C) is correct.