"The infinite dihedral groupD∞ is generated (as a subgroup of the group SR of
bijections : R ! R), by the translation t(x) = x + 1 and the reflection s(x) = −x of
the real line. Work out its elements, and find the orbit and the stabilizer of each of
the points 1, 1/2, 1/3"
I have to deal with that problem but i have no idea that how i am going to use that points. I once read that ts(x)= -x+1 has reflection in the point 1/2 but i dont know how.
Anyone to explain?
Originally Posted by dreamon
first see thatand thus
which implies that
for all
therefore
for the second part of your question, to find the orbit of
we need to findfor any
we have:
so we can write
thus
therefore the orbit of
is the set
in order to find he stabilizer of
such that
that will depend on
for example, if
then we need to
solveif
then we have
and if
then
so the stabilizer of
is
forwe need to solve
which can be written as
now if
and if
so the stabilizer of
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