Thread: Group theory

Having some trouble figuring this out - any help at all would be greatly appreciated! Thanks

S the group of bijective applications R - > R. x, y elements of S. For every real number t, x(t) = t +1, y(t) = 2t.
G is a sub-group of S generated by {x,y}.

1. For every integer n>=1, xn = y^(-n)xy^(n).
Calculate xn(t) and
show that (xn)^2 = xn-1

2. Let Hn be a subgroup of G generated by xn Show that Hn is a subgroup of Hn+1 and that Hn is not equal to Hn+1

Originally Posted by rmangan

S the group of bijective applications R - > R. x, y elements of S. For every real number t, x(t) = t +1, y(t) = 2t.
G is a sub-group of S generated by {x,y}.

1. For every integer n>=1, xn = y^(-n)xy^(n).
Calculate xn(t) and
show that (xn)^2 = xn-1

2. Let Hn be a subgroup of G generated by xn Show that Hn is a subgroup of Hn+1 and that Hn is not equal to Hn+1

The elements of all bijections on are functions defined by and . Now . Also, .
Thus, .
We see that and so .

Wii see that because .
Thus, and can you show now ?