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Ok, so we know that:OK, this following paragraph is where I'm stuck: Since K is normal, yxy^-1 is an element of K, a power of x, say x^i, where i is between 1 and 7. So the elements x and y satisfy the relations x^7=1, y^3=1, yx=(x^i)y. OK, this makes sense. But, says Artin, the relation y^3 restricts the possible exponents i:
x=(y^3)(x)(y^-3) = (y^2)(x^i)(y^-2) = y(x^(i^2))(y^-1)=x^(i^3)).