Unformatted text preview: 1 ) 1 . . . x π ( n + 1 ) n + 1 , π ( i ) ∈ Z , and thus g < x n + 1 > = x π ( 1 ) 1 . . . x π ( n + 1 ) n + 1 < x n + 1 > = x π ( 1 ) 1 . . . x π ( 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 subgroup of G/ < x n + 1 > (using the fact that H is a subgroup of G ). By inductive hypothesis H < x n + 1 > is ﬁnitely 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 π ( 1 ) 1 . . . y π ( k ) k x m n + 1 , with π ( i ) , m ∈ Z . Hence H is ﬁnitely generated. ±...
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 '08
 YEE
 Math, Normal subgroup, Trigraph, Cyclic group, Coset

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