Let g be arbitrary group and let a,b. Belongs to g be two distinct elements.the which of the following statement is true .order of ab equals to order of ba?

Question

Answer

Statement:

The statement 'The order of ab equals the order of ba' is not necessarily true.

Explanation:

In a group, the order of an element refers to the smallest positive integer n such that a^n = e, where e is the identity element of the group.

Counterexample:

Let's consider a counterexample to demonstrate that the statement is not always true.

Counterexample:

Let g be the group of integers modulo 4 under addition (g = Z4). The elements of this group are {0, 1, 2, 3}.

Consider a = 1 and b = 2. The order of a is 4 because 1 + 1 + 1 + 1 = 0 (mod 4), and the order of b is also 4 because 2 + 2 + 2 + 2 = 0 (mod 4).

Now, let's compute the products ab and ba:

ab = 1 + 2 = 3 (mod 4)

ba = 2 + 1 = 3 (mod 4)

Both ab and ba are equal to 3 (mod 4).

Therefore, in this case, the order of ab is 4, but the order of ba is 1, which is not equal.

Conclusion:

Based on the counterexample, we can conclude that the statement 'The order of ab equals the order of ba' is not always true for an arbitrary group g and two distinct elements a and b in that group. It is important to consider specific examples and properties of the group to determine the relationship between the orders of ab and ba.

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