If x and y are elements in a group G and if x5 = y3 = e, where e is the…
2025
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
- A.
y2xy2x4
- B.
yxy2x3
- C.
yx6y6x3
- D.
x4y2x2y
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Correct answer: B
Key idea: In a group, the inverse of a product is found by reversing the order and taking the inverse of each factor.
Given x5 = e, so x⁻¹ = x4. Hence (x2)⁻¹ = x3 and (x4)⁻¹ = x.
Given y3 = e, so y⁻¹ = y2. Hence (y2)⁻¹ = y.
(x2yx4y2)⁻¹ = (y2)⁻¹(x4)⁻¹y⁻¹(x2)⁻¹ = yxy2x3.
Therefore, the correct answer is yxy2x3.
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