While we're on the subject of the meaning of
infinite paths in N, would
anyone like to comment on the validity/invalidity
of the following argument?
I was looking at the group G defined by the
quaternion-like relations
ijk = l
jkl = i
kli = j
lij = k
If you define two "permutations of N"
t = (1)(2 3)(4 5) (6 7)....
u = (1 2)(3 4)(5 6)...
their product is a sort of
infinite-order "cycle" in N
tu = (1 3 5 7 9 11 13 15 17 ...... 16
14 12 10 8 6 4 2).
Now consider the mapping f
f(i) = t
f(j) = t
f(k)= u
f(l) = u
The mapping f seems to be a homomorphism from G
into the group of
permutations of N that respects the 4 group
relations I started out with
(when you check the relations, the involutions t
and u cancel each other out).
So in particular the element f(ik) = tu has
infinite order, so G is infinite.
I'm sure this is nothing new, but I was happy to
find it (even more happy if it is valid).