Briefly, this is because every-orbit admits representatives of the form, and the representative is determined only up to left multiplication by an element of.
12.
It is possible to restate the characteristic property of a cancellative element in terms of a property held by the corresponding left multiplication and right multiplication maps defined by,.
13.
If one takes " F " & prime; to be " G " with the action of left multiplication then one obtains the associated principal bundle.
14.
Remark : Some intuition for the mean ergodic theorem can be developed by considering the case where complex numbers of unit length are regarded as unitary transformations on the complex plane ( by left multiplication ).
15.
Two sets of coordinates represent the same point if they are'proportional'by a left multiplication by a non-zero quaternion " c "; that is, we identify all the
16.
The integral curves or fibers respectively are certain " pairwise linked " great circles, the orbits in the space of unit norm quaternions under left multiplication by a given unit quaternion of unit norm.
17.
The natural action of group " G " 1 above and its action on itself ( via left multiplication ) are not equivalent as the natural action has fixed points and the second action does not.
18.
Every topological group can be viewed as a uniform space in two ways; the " left uniformity " turns all left multiplications into uniformly continuous maps while the " right uniformity " turns all right multiplications into uniformly continuous maps.
19.
:There is a natural action of R on itself by left multiplication; this action descends to an action of R on R / I by left multiplication . ( Explicitly, r acts on x + I by sending it to rx + I, using the additive notation for cosets.
20.
:There is a natural action of R on itself by left multiplication; this action descends to an action of R on R / I by left multiplication . ( Explicitly, r acts on x + I by sending it to rx + I, using the additive notation for cosets.
