Difference between revisions of "Abelian Group"

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)

@@ Line 1: / Line 1: @@
 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
+(A) 2,2,3
-(B)3,3,3
+(B) 3,3,3
-(C)2,2,4
+(C) 2,2,4
-'''(D)2,3,4'''
+'''(D) 2,4,4'''
-===Solution===
+==={{Template:Author|Arjun Suresh|{{arjunweb}} }}===
+[https://en.wikipedia.org/wiki/Abelian_group Abelian_group]
+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.
+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.
 {| class="wikitable" style="text-align: center;background-color: #ffffff;" width="35%"
@@ Line 28: / Line 47: @@
 |a
 |e
-|b
+|c
 |b
 |-
@@ Line 34: / Line 53: @@
 |b
 |c
-|e
+|a
 |e
 |-
@@ Line 44: / Line 63: @@
 |}
-a and b have order 2(a * a = e and b * b = e). c has order 4 (since c * c = a and a * a = e)
+ a * a = e => order(a) = 2
+ b * b * b * b = a * b * b = c * b = e => order(b) = 4
-<div class="fb-like"  data-layout="standard" data-action="like"  data-share="true"></div>
+ c * c * c * c = a * c * c = b * c = e => order(c) = 4
-<div class="fb-share-button"  data-type="button_count"></div>
-<disqus/>
+So, the answer is 2,4,4. (2,2,2 is another possibility)
+{{Template:FBD}}
-[[Category:Other Questions]]
+[[Category: Non-GATE Questions from Graph Theory]]
-[[Category: Algorithms & Data Structures]]

Difference between revisions of "Abelian Group"

Latest revision as of 11:37, 15 July 2014

Solution by Arjun Suresh

Solution[edit]