Intuitive interpretation of the commutator of two elements of a group. Commutator subgroups (or derived groups). Commutators of permutations.

SolitaryRoad.com

Website owner:  James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

Intuitive interpretation of the commutator a^-1 b^-1ab of two elements of a group. Commutator subgroups (or derived groups). Commutators of permutations. Proper and improper motions of space.

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.

Left and right quotients in a group. In ordinary algebra the quotient of b divided by a is given by the solution of the equation ax = b, i.e. x = a^-1b. In a group, because multiplication need not be commutative, ab ≠ ba, the solution of ax = b (x = a^-1b), is in general different from the solution of xa = b (x = ba^-1), so we must speak about right quotients and left quotients. The solution of ax = b or x = a^-1b is called the right quotient and the solution of xa = b or x = ba^-1 is called the left quotient of b divided by a.

Def. Commutator of two elements of a group. The commutator of two elements a and b of a group is the element c = a^-1 b^-1ab.

The commutator c = a^-1 b^-1ab of elements a and b can be viewed as the quotient on “division” of the conjugate transformation b^-1ab by a i.e. the solution of the equation ax = b^-1ab.

For intuitive insight, consider an example involving motions of the plane. If A is a translation of the plane, then a conjugate transformation B^-1AB of A by B is also a translation and the quotient A^-1B^-1AB of B^-1AB by A is also a translation. Thus the commutator of a translation and an arbitrary motion of space is a translation.

Let A be a rotation around a particular point O by an angle φ and let B be a rotation or translation. Then the conjugate transformation B^-1AB is again a rotation by the angle φ, but around another point Oʹ. Therefore the commutator c(A, B) = A^-1B^-1AB is, in this case, the product of a rotation around O by an angle -φ (arising from the A^-1 factor) and a rotation around Oʹ through the positive angle φ due to the conjugate transformation B^-1AB. See Fig. 1. In other words, the square OPQR is rotated about point O, taking it from position I into position II (OʹPʹQʹRʹ), by the A^-1 multiplication and then it is rotated about point Oʹ into position III (OʹʹPʹʹQʹʹRʹʹ) by the conjugate transformation B^-1AB. It is clear from the figure that the final result is a simple translation of square OPQR — moving it in the direction of the vector O Oʹʹ by the distance O Oʹʹ .

We have thus arrived at the interesting fact that for a plane the commutator of any two motions of the first kind (whether translation or rotation) is a parallel shift (i.e. a translation) or the identity transformation.

Commutator subgroups (or derived groups). The subgroup generated by the commutators of all elements of a group G is called the commutator subgroup or derived group of G. The derived group of G consists precisely of those elements that can be represented in the form of products of commutators. Since in a plane the commutator of any two motions of the first kind is a parallel shift, products of parallel shifts are again parallel shifts, we can say that the derived group of the group of motions of the first kind consists only of parallel shifts.

The derived group of an Abelian group consists only of the identity transformation. This fact follows directly from the fact that in an Abelian group ab = ba since a^-1b^-1ab = (ba)^-1ab = (ab)^-1ab = e.

The derived group of a group G is often denoted by Gʹ. The derived group of the derived group is called the second derived group of G and is denoted by Gʹʹ. By repeating this process we can define the derived group of arbitrary order of a group G.

Commutators of permutations.

Theorem. The commutator c(A, B) = A^-1B^-1AB of any two permutations A, B is always an even permutation.

Proof. Let G be the symmetric group of all permutations of the numbers 1, 2, ... , n. The permutation AB, BA and consequently also (BA)^-1, always have the same parity. As a consequence the commutator c(A, B) = (BA)^-1AB, as the product of permutations of equal parity, is an even permutation.

The derived group of the symmetric group S_n consists of even permutations only. It is easily shown that it in fact coincides with the whole alternating group A_n.

If among the derived groups of a group G at least one (and hence all subsequent ones) consists of the identity transformation only, then the group is called solvable. This name has risen in the theory of equations, where solvability of a group corresponds to solvability of an equation by radicals. The group of motions of the first kind in a plane is solvable, because its second derived group is the identity. The symmetric groups of degree 2, 3, and 4 are solvable, because their first, second, and third derived groups, respectively, are the identity. On the other hand, the symmetric groups of degree 5 and higher are not solvable because it can be shown that their second derived group coincide with the first and is different from the identity.

Portions excerpted from Mathematics, Its Content, Methods and Meaning. Chap. XX

References.

Mathematics, Its Content, Methods and Meaning. Chap. XX