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'''(D) 2,4,4'''
 
'''(D) 2,4,4'''
  
===Solution===
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==={{Template:Author|Arjun Suresh|{{arjunweb}} }}===
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[https://en.wikipedia.org/wiki/Abelian_group Abelian_group]
  
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As a consequence of [https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory) Lagrange's theorem], the order of every element of a group divides the order of the group. Hence, 3 cannot be the order of any element in the group.
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Now, consider 2,2,4. If it has to hold then,
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a * a = e
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b * b = e and
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c * c = a
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=> a * c = b and
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b * c = e (to get $c^4 = e$)
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But then, the associativity property of
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[(a * c) * b] = [a * (c * b)]
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fails as (a * c) * b = e and a * (c * b) = a. Hence, 2,2,4 is not the answer.
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  c * c * c * c = a * c * c = b * c = e => order(c) = 4
 
  c * c * c * c = a * c * c = b * c = e => order(c) = 4
  
<div class="fb-like"  data-layout="standard" data-action="like"  data-share="true"></div>
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So, the answer is 2,4,4. (2,2,2 is another possibility)
 
 
 
 
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<disqus/>
 
  
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{{Template:FBD}}
  
[[Category:Other Questions]]
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[[Category: Non-GATE Questions from Graph Theory]]
[[Category: Algorithms & Data Structures]]
 

Latest revision as of 11:37, 15 July 2014

Let <math>G\{e,a,b,c\}</math> be an abelian group with <math>'e'</math> as an identity element. The order of the other elements are:

(A) 2,2,3

(B) 3,3,3

(C) 2,2,4

(D) 2,4,4

Solution by Arjun Suresh

Abelian_group

As a consequence of Lagrange's theorem, the order of every element of a group divides the order of the group. Hence, 3 cannot be the order of any element in the group.

Now, consider 2,2,4. If it has to hold then,

a * a = e

b * b = e and

c * c = a

=> a * c = b and

b * c = e (to get $c^4 = e$)

But then, the associativity property of

[(a * c) * b] = [a * (c * b)] 

fails as (a * c) * b = e and a * (c * b) = a. Hence, 2,2,4 is not the answer.

* e a b c
e e a b c
a a e c b
b b c a e
c c b e a
a * a = e => order(a) = 2
b * b * b * b = a * b * b = c * b = e => order(b) = 4
c * c * c * c = a * c * c = b * c = e => order(c) = 4

So, the answer is 2,4,4. (2,2,2 is another possibility)




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Let <math>G\{e,a,b,c\}</math> be an abelian group with <math>'e'</math> as an identity element. The order of the other elements are:

(A) 2,2,3

(B) 3,3,3

(C) 2,2,4

(D) 2,4,4

Solution[edit]

* e a b c
e e a b c
a a e c b
b b c a e
c c b e a
a * a = e => order(a) = 2
b * b * b * b = a * b * b = c * b = e => order(b) = 4
c * c * c * c = a * c * c = b * c = e => order(c) = 4



blog comments powered by Disqus