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Return to Abelian Group.
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
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)
GATECSE