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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 | b | d | 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
An element is a generator for a cyclic group if on repeated applications of it upon itself, it can generate all elements of group.
For example here : $a*a = a$, then $(a*a)*a = a*a = a$, and so on. Here we see that no matter how many times we apply $a$ on itself, we can't generate any other element except $a$, so $a$ is not a generator.
Now for $b$, $b*b = a$. Then $(b*b)*b = a*b = b$. Then $(b*b*b)*b = b*b = a$, and so on. Here again we see that we can only generate $a$ and $b$ on repeated application of $b$ on itself. So it is not a generator.
Now for $c$, $c*c = b$. Then $(c*c)*c = b*c = d$. Then $(c*c*c)*c = d*c = a$. Then $(c*c*c*c)*c = a*c = c$. We see that we have generated all elements of group, so $c$ is a generator.
For $d$, $d*d = b$. Then $(d*d)*d = b*d = c$. Then $(d*d*d)*d = c*d = a$. Then $(d*d*d*d)*d = a*d = d$. We have generated all elements of group from $d$, so $d$ is a generator.
Thus $c$ and $d$ are generators. So option (C) is correct.
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 | b | d | 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
An element is a generator for a cyclic group if on repeated applications of it upon itself, it can generate all elements of group.
For example here : $a*a = a$, then $(a*a)*a = a*a = a$, and so on. Here we see that no matter how many times we apply $a$ on itself, we can't generate any other element except $a$, so $a$ is not a generator.
Now for $b$, $b*b = a$. Then $(b*b)*b = a*b = b$. Then $(b*b*b)*b = b*b = a$, and so on. Here again we see that we can only generate $a$ and $b$ on repeated application of $b$ on itself. So it is not a generator.
Now for $c$, $c*c = b$. Then $(c*c)*c = b*c = d$. Then $(c*c*c)*c = d*c = a$. Then $(c*c*c*c)*c = a*c = c$. We see that we have generated all elements of group, so $c$ is a generator.
For $d$, $d*d = b$. Then $(d*d)*d = b*d = c$. Then $(d*d*d)*d = c*d = a$. Then $(d*d*d*d)*d = a*d = d$. We have generated all elements of group from $d$, so $d$ is a generator.
Thus $c$ and $d$ are generators. So option (C) is correct.
GATECSE