One-parameter Group - Discussion

Discussion

That is, we start knowing only that

φ (s + t) = φ(s)φ(t)

where s, t are the 'parameters' of group elements in G. We may have

φ(s) = e, the identity element in G,

for some s ≠ 0. This happens for example if G is the unit circle and

φ(s) = eis.

In that case the kernel of φ consists of the integer multiples of 2π.

The action of a one-parameter group on a set is known as a flow.

A technical complication is that φ(R) as a subspace of G may carry a topology that is coarser than that on R; this may happen in cases where φ is injective. Think for example of the case where G is a torus T, and φ is constructed by winding a straight line round T at an irrational slope.

Therefore a one-parameter group or one-parameter subgroup has to be distinguished from a group or subgroup itself, for the three reasons

1. it has a definite parametrization,
2. the group homomorphism may not be injective, and
3. the induced topology may not be the standard one of the real line.