If x and y are elements in a group G and if x5 = y3 = e, where e is the identity of G, then the inverse of x2yx4y2 must be

The correct answer is: Option 2

Key Points

Concept:

The inverse of a product of elements in a group is the reverse product of their inverses. Given and , we use:
1. (since ).
2. (since ).

Given:

Element to invert:
Conditions: (identity).

Steps:

1. Apply the inverse property in reverse order:
.

2. Compute each inverse:
- (since ).
- (since ).
- .
- (since ).

3. Substitute and simplify:
.

Verification:
Multiply by :
- , , , .
Result: (identity), confirming correctness.

Final Answer: ✅ Option 2 ()