Let G be a group having elements a& b st O(a) =4, O(b)=2 and a^3b =ba . Then O(a b) is equal to?

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Answer

And b such that ab = ba = e, where e is the identity element of G. Then, the element a and b are inverses of each other.

To prove this, we need to show that ab = ba = e implies that a is the inverse of b and b is the inverse of a.

First, let's prove that a is the inverse of b. Since ab = e, we have that a is a right inverse of b. Now, we need to show that a is also a left inverse of b, which means we need to show that ba = e.

To prove ba = e, we can multiply both sides of the equation ab = e by b on the left:

ba = (ab)b = a(bb) = ae = ae = a.

Therefore, we have ba = e, which means a is a left inverse of b.

Since a is both a left and a right inverse of b, we can conclude that a is the inverse of b.

Similarly, we can prove that b is the inverse of a using the same steps.

Hence, the element a and b in group G are inverses of each other.

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