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Which one of the following choices is correct?
 
Which one of the following choices is correct?
<br>
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<b>(A) </b>a, b are generators
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(A) $a, b$ are generators
&nbsp;
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<b>(B) </b>b, c are generators
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(B) $b, c$ are generators
<br>
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<b>(C) </b>c, d are generators
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(C) $c, d$ are generators
&nbsp;
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<b>(D) </b>d, a are generators
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(D) $d, a$ are generators
 
==={{Template:Author|Happy Mittal|{{mittalweb}} }}===
 
==={{Template:Author|Happy Mittal|{{mittalweb}} }}===
  
 
An element is a generator for a cyclic group if on repeated applications of it upon itself, it can generate all  
 
An element is a generator for a cyclic group if on repeated applications of it upon itself, it can generate all  
 
elements of group.
 
elements of group.
<br>
+
 
For example here : a*a = <b>a</b>, then (a*a)*a = a*a = <b>a</b>, and so on. Here we see that no matter how many times we apply a on itself,  
+
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.
+
we can't generate any other element except $a$, so $a$ is not a generator.
<br>
+
Now for b, b*b = <b>a</b>. Then (b*b)*b = a*b = <b>b</b>. Then (b*b*b)*b = b*b = <b>a</b>, and so on. Here again we see that we can only generate a and b
+
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.
+
on repeated application of $b$ on itself. So it is not a generator.
<br>
+
Now for c, c*c = <b>b</b>. Then (c*c)*c = b*c = <b>d</b>. Then (c*c*c)*c = d*c = <b>a</b>. Then (c*c*c*c)*c = a*c = <b>c</b>. So we see that we have generated
+
Now for $c$, $c*c = b$. Then $(c*c)*c = b*c = d$. Then $(c*c*c)*c = d*c = a$. Then $(c*c*c*c)*c = a*c = c$. So we see that we have generated
all elements of group. So c is a generator.
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all elements of group. So $c$ is a generator.
<br>
+
For d, d*d = <b>b</b>. Then (d*d)*d = b*d = <b>c</b>. Then (d*d*d)*d = c*d = <b>a</b>. Then (d*d*d*d)*d = a*d = <b>d</b>. So we have generated all elements
+
For $d$, $d*d = b$. Then $(d*d)*d = b*d = c$. Then $(d*d*d)*d = c*d = a$. Then $(d*d*d*d)*d = a*d = d$. So we have generated all elements
of group from d, so d is a generator.
+
of group from $d$, so $d$ is a generator.
<br>
+
So c and d are generators. So option <b>(C)</b> is correct.
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So $c$ and $d$ are generators. So option <b>(C)</b> is correct.
  
  

Revision as of 12:35, 14 July 2014

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

Solution by Happy Mittal

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 $(c*c*c*c)*c = a*c = c$. So 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 $(d*d*d*d)*d = a*d = d$. So we have generated all elements of group from $d$, so $d$ is a generator.

So $c$ and $d$ are generators. So option (C) is correct.




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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

Solution by Happy Mittal[edit]

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 $(c*c*c*c)*c = a*c = c$. So 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 $(d*d*d*d)*d = a*d = d$. So we have generated all elements of group from $d$, so $d$ is a generator.

So $c$ and $d$ are generators. So option (C) is correct.




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