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Let $G$ be a group with identity element $e$. If $x$ and $y$ are elements in $G$ satisfying $x^5y^3=x^8y^5=e$, then which of the following conditions is true?

  1. $x=e, \: y=e$
  2. $x\neq e, \: y=e$
  3. $x=e, \: y \neq e$
  4. $x\neq e, \: y \neq e$
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1 Answer

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Here 

x^5 ^y3 = x^8 y^5 = e   (given)

by cancellation property-

x^5 ^y3 = x^8 y^5

x^3 y^2 = e   -------------(1)

CASE 1-    either x^3 can be the inverse of y^2, giving identity element e

CASE 2-    either y^2 can be the inverse of x^3, giving identity element e

CASE 3-    if not case 1 and 2, then both have to be identity element e to satisfy (1)

hence option A is correct

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