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
- it has a definite parametrization,
- the group homomorphism may not be injective, and
- the induced topology may not be the standard one of the real line.
Read more about this topic: One-parameter Group
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