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A question of Mazurov on groups of exponent dividing 12

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Mazurov asked whether a group of exponent dividing 12, which is generated by x, y and z subject to the relations x 3 = y 2 = z 2 = ( x y ) 3 = ( y z ) 3 = 1 , has order at most 12. We show that if such a group is finite, then the answer is yes. ... ...