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
Famous quotes containing the word discussion:
“It was heady stuff, recognizing ourselves as an oppressed class, but the level of discussion was poor. We explained systemic discrimination, and men looked prettily confused and said: “But, I like women.””
—Jane O’Reilly, U.S. feminist and humorist. The Girl I Left Behind, ch. 2 (1980)
“My companion and I, having a minute’s discussion on some point of ancient history, were amused by the attitude which the Indian, who could not tell what we were talking about, assumed. He constituted himself umpire, and, judging by our air and gesture, he very seriously remarked from time to time, “you beat,” or “he beat.””
—Henry David Thoreau (1817–1862)
“This is certainly not the place for a discourse about what festivals are for. Discussions on this theme were plentiful during that phase of preparation and on the whole were fruitless. My experience is that discussion is fruitless. What sets forth and demonstrates is the sight of events in action, is living through these events and understanding them.”
—Doris Lessing (b. 1919)