(7 intermediate revisions by the same user not shown)
Line 9: Line 9:
 
'''(D) 2,4,4'''
 
'''(D) 2,4,4'''
  
===Solution===
+
==={{Template:Author|Arjun Suresh|{{arjunweb}} }}===
 
[https://en.wikipedia.org/wiki/Abelian_group Abelian_group]
 
[https://en.wikipedia.org/wiki/Abelian_group Abelian_group]
  
Line 15: Line 15:
  
 
Now, consider 2,2,4. If it has to hold then,  
 
Now, consider 2,2,4. If it has to hold then,  
a * a = e
+
 
b * b = e and
+
a * a = e
c * c = a
+
 
=> a * c = b and
+
b * b = e and
b * c = e (to get $c^4 = e$)
+
 
 +
c * c = a
 +
 
 +
=> a * c = b and
 +
 
 +
b * c = e (to get $c^4 = e$)
 +
 
 
But then, the associativity property of  
 
But then, the associativity property of  
 
  [(a * c) * b] = [a * (c * b)]  
 
  [(a * c) * b] = [a * (c * b)]  
Line 62: Line 68:
  
 
So, the answer is 2,4,4. (2,2,2 is another possibility)
 
So, the answer is 2,4,4. (2,2,2 is another possibility)
<div class="fb-like"  data-layout="standard" data-action="like"  data-share="true"></div>
 
 
 
<div class="fb-share-button"  data-type="button_count"></div>
 
 
 
 
<disqus/>
 
  
 +
{{Template:FBD}}
  
[[Category:Other Questions]]
+
[[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)




blog comments powered by Disqus

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]

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)



blog comments powered by Disqus