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**Unformatted text preview: **1 ) 1 . . . x ( n + 1 ) n + 1 , ( i ) Z , and thus g < x n + 1 > = x ( 1 ) 1 . . . x ( n + 1 ) n + 1 < x n + 1 > = x ( 1 ) 1 . . . x ( n ) n < x n + 1 > . Now consider the set H < x n + 1 > = { h < x n + 1 > : h H } . It is not hard to see that this is a sub-group of G/ < x n + 1 > (using the fact that H is a subgroup of G ). By inductive hypothesis H < x n + 1 > is nitely generated, say by { y 1 , . . . y k } . Given h H , we have that h h < x n + 1 > . Thus h can be written as y ( 1 ) 1 . . . y ( k ) k x m n + 1 , with ( i ) , m Z . Hence H is nitely generated....

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