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Motions of space. Intuitive meaning of the concept of a
conjugate transformation B^{-1}AB of a transformation A

Motions of space. The possible motions of space (three dimensional space) are classified as proper motions (or motions of the first kind) and improper motions (or motions of the second kind).

Proper motions (motions of the first kind) are translations, rotations, and spiral motions. Improper motions (motions of the second kind) are all other motions, including reflections.

In a plane the proper motions are translations and rotations. The improper motions are all other motions including reflections.

Let us note that the set of all possible translations in the plane meet the axiomatic requirements for a group. Collectively they form a group. The same can be said for the set of all possible rotations.

Conjugate transformations. The transformation B^{-1}AB is said to be obtained from A by
transforming A by B. The element C = B^{-1}AB is said to be conjugate to A by means of B.

What is the intuitive meaning of the transformation B^{-1}AB? The answer lies in the following
theorem:

Theorem 1. If the image of *m* is *n* under transformation A (i.e. *m*A = *n*) then the image of *m*B
is *n*B under the transformation C = B^{-1}AB. That is, if *m*A = *n*, then *m*BC =
*n*B, or equivalently,

*m*B(B^{-1}AB) = *n*B

Proof. (mB)B^{-1}AB = m(BB^{-1})AB =
mAB = nB

Note here that C is a conjugate of A by
means of B. In general, the transform
x^{-1}ax decomposes the members of a
group into k disjoint equivalence
classes where members of the same
class are said to be conjugate to each
other. What is the common property
that distinguishes the members of a
particular equivalence class? What
property is it that makes members of a particular conjugate class conjugate to each other?

1. Intuitive meaning of the concept of a conjugate transformation B^{-1}AB of a
transformation A for motions of the plane. There are two motions of the plane:
translation and rotation. Let R represent a rotation of the plane around the origin O through the
angle φ. Let S be a translation of the plane by an amount a in the positive x direction and let Q
be the point into which the origin O is carried by the translation. See Figure 1. Let C = S^{-1}RS be
the conjugate of R by means of S. We wish to show that if the image of m is n under
transformation R (rotation) about origin O (i.e. mR = n) then the point mS is carried into nS by
the transformation C = S^{-1}RS. In other words, we wish to show that if mR = n, then (mS)C = nS.
That it does is obvious from

1) (mS)C = (mS)(S^{-1}RS) = (mR)S = nS

since (mS)S^{-1} = m and (mR) = n. Each of these operations can be followed, a step at a time,
from Fig. 1. We can see that the transformation C = S^{-1}RS represents a rotation about the point
Q by the same angle φ as the transformation R. Following 1) a step at a time, we see that S^{-1}
carries mS into m, R carries m into n, and S carries n into nS. Thus S^{-1}RS carries mS into nS.

Transforms of the motions of the plane

The transform of an element a of a group G by an element x is the element b = x^{-1}ax. The
element b is said to be conjugate to a by means of x.

Transform of a rotation by a translation. Let R_{O} be a rotation of a plane around a point O
through the angle φ. Let S be a translation of the plane. Then the transform C = S^{-1} R_{O}S of
rotation R_{O} by translation S is a rotation of the plane around another point Oʹ in the plane by that
same angle φ, where Oʹis the point into which O is carried by the translation S.

Transform of a translation by a rotation. If a translation of the plane characterized by the
vector A is transformed by means of a rotation R_{O} by the angle φ, we obtain again a translation of
the plane, characterized by a different vector B.

Transform of a translation by a translation. The transform of a translation by a translation is a translation.

Transform of a rotation by a rotation. The transform of a rotation by a rotation is a rotation.

Thus the conjugate of a rotation by means of a translation is a rotation; the conjugate of a translation by means of a rotation is a translation; the conjugate of a translation by means of a translation is a translation; the conjugate of a rotation by means of a rotation is a rotation.

2. Intuitive meaning of the concept of a conjugate transformation B^{-1}AB of a
transformation A for the case of permutation groups. Just what is the intuitive
meaning of the transformation C = B^{-1}AB when A and B represent elements of permutation
groups?

Some notation and conventions. We will denote transformations of a finite set M = {m_{1},
m_{2}, ... , m_{n}} into itself, transformations such as

by capital letters such as A, B, C, etc. In addition, we will denote the image of an arbitrary
element *m* in M by the notation *m*A. For example, if

then 4A = 3, 2A = 4, 1A = 2, 3A = 1.

Let

then

where

represent the images of a_{1}, a_{2, ... , }a_{n} respectively under the transformation B i.e.

If we examine the matrix

we note that the upper row is obtained by subjecting all the elements in the upper row of A to the
transformation specified by B and that the lower row is obtained by subjecting all the elements in
the lower row of A to the transformation specified by B (in other words, the upper and lower
rows of B^{-1}AB are the images of the upper and lower rows of A under the transformation B).

Example. Let

then

Let us now ask a question: How does the above result relate to Theorem 1? Well, the* m* of
Theorem 1 corresponds to the upper row of A and the *n* of Theorem 1 corresponds to the lower
row of A. Transformation

is viewed as transforming the ordered set (1, 2, ... , n) into the ordered set (a_{1}, a_{2, ... , }a_{n}). The upper
row of B^{-1}AB then consists of the image of *m* under B and the lower row consists of the image of
*n* under B. Thus the transformation B^{-1}AB does then constitute a transformation that carries *m*B
into *n*B. So the above result does follow directly from Theorem 1.

In the case where permutations A and B are expressed in cyclic notation the above result can be expressed as follows:

Theorem 2. Let A be a cyclic permutation A = (a_{1}, a_{2, ... , }a_{n}). Then the conjugate
transformation C = B^{-1}AB is given by C = (a_{1}B, a_{2}B, ... , a_{n}B).

In other words, the cyclic representation of C = B^{-1}AB is obtained from the cyclic representation
of A by replacing each element in the cyclic representation of A by its image under B.

A direct consequence of Theorem 2 is the following corollary:

Corollary. The conjugate of any cycle in a permutation group is another cycle of the same length.

References

Mathematics, Its Content, Methods and Meaning. Vol. 3

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