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ALGEBRA
By Larry C. Grove Dover Publications, Inc.
Copyright © 1983 Larry C. Grove
All rights reserved.
ISBN: 978-0-486-14213-5
CHAPTER 1
Groups
1. GROUPS, SUBGROUPS, AND HOMOMORPHISMS
A nonempty set with an associative binary operation is called a semigroup, and a semigroup S having an identity element 1 such that 1x = x1 = x for all x [member of] S is called a monoid. Most of the algebraic systems discussed herein will be semigroups or monoids, but almost always with further requirements imposed, so the semigroup or monoid aspect will seldom be explicitly emphasized.
One trivial consequence of the definition of a monoid deserves mention.
Proposition 1.1. The identity element of a monoid S is unique.
Proof. Suppose 1 and e are identities in S. Then 1 = 1e = e.
A group is a set G with an associative binary operation (usually called multiplication) and an identity element 1 satisfying the further requirement that for each x [member of] G there is an inverse element y [member of] G such that xy = yx = 1.
Proposition 1.2. If G is a group and x [member of] G, then x has a unique inverse element.
Proof. Let y and z be inverses for x. Then
y = y1 = y(xz) = (yx)z = 1z = z.
The unique inverse for x [member of] G is denoted by x-1. Note that (x-1)1 = x.
Proposition 1.3. If G is a group and x, y [member of] G, then (xy)-1 = y-1x-1.
Proof
(xy)(y-1 x-1) = ((xy)y-1)x-1 = (x(yy-1))x-1 = (x1)x-1 = xx-1 = 1, and similarly (y-1x-1)(xy) = 1.
As Coxeter [7] has pointed out, the "reversal of order" in Proposition 1.3 becomes clear when we think of the operations of putting on our shoes and socks.
If the binary operation of a group G is written as addition, then the identity element is commonly denoted by 0 rather than 1, and the inverse of x by — x rather than x-1. It is customary to use additive notation only if x + y = y + x for all x, y [member of] G.
In general, a group G (multiplicative again) is called abelian (or commutative) if xy = yx for all x, y [member of] G.
We write x0 = 1, x1 = x, x2 = xx, and in general x = xn-1x for 1 ≤ n [member of] [member of] Z. Define x-n = (x-1)n, again for 1 ≤ n [member of] Z. It is easy to verify by induction that the usual laws of exponents hold in any group, viz.,
xmxn = xm+n and (xm)n = xmn
for all x [member of] G, all m, n [member of] Z. The additive analog of xn is nx, so the additive analogs of the laws of exponents are mx + nx = (m + n)x and n(mx) = (mn)x.
Exercise 1.1. Verify the laws of exponents for groups.
EXAMPLES
1. Let G = {1, — 1} [subset or equal to] R, with multiplication as usual. Then G is a group.
2. Let G = Z, Q, R, or C, with the usual binary operation of addition. Then G is a group.
3. Let G = Q\{0}, the set of nonzero rational numbers, under multiplication. Then G is a group. Similarly this holds for R\{0} and C\{0}, but not for Z\{0}. (Why?)
4. Let S be a nonempty set. A permutation of S (sometimes called a bijection of S) is a 1—1 function φ from S onto S. Let G be the set of all permutations of S. If φ, θ [member of] G, we define φθ to be their composition product, i.e., φθ(s) = φ(θ(s)) for all s [member of] S. Composition is a binary operation on G (verify), and it is associative, for if φ, θ, σ [member of] G and s [member of] S, then
(φ(θσ))(s) = φ(θσ(s)) = φ [θ(σ(s))],
and
((φθ)σ)(s) = φθ(σ(s)) = φ[θ(σ(s))].
G has an identity element, the permutation 1 = 1s defined by 1(s) = s, all s [member of] S, and each φ [member of] G has an inverse φ-1 defined by φ-1(s1) = s2 if and only if φ(s2) = s1 (there are a few details to be verified). Thus G is a group; we write G = Perm(S). This example is of considerable importance and will be pursued much further.
5. As a special case of the preceding example take S = {1, 2, 3, ..., n}. The group G of all permutations of S is called the symmetric group on n letters and is denoted by G = Sn. If φ [member of] Sn, it is convenient to display the function φ explicitly in the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
For example, if n = 3, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the permutation that maps 1 to 2, 2 to 3, and 3 to 1. The notation makes it quite simple to carry out explicit computations of the composition product. Suppose, for example, that n = 3 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Note from the definition of φθ in Example 4 that θ acts first and φ second. Thus θ maps 1 to 3 and φ then maps 3 to 1, and so the composite φθ maps 1 to 1. Similarly, φθ maps 2 to 3 and maps 3 to 2. Thus
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Observe that
so S3 is not an abelian group. It is easy to see that Sn is, likewise, not abelian for any n > 3, although S1 and S2 are abelian.
6. Let T be an equilateral triangle in the plane with center O. Let D3 denote the set of symmetries of T, i.e., distance-preserving functions from the plane onto itself that carry T onto T (as a set of points). The elements of D3 are called congruences of the triangle T in plane geometry. With composition as the binary operation, D3 is a group. Let us list its elements explicitly. There is, of course, the identity function 1, with 1(x) = x for all x in the plane. There are two counterclockwise rotations, φ1 and φ2, about O as center through angles of 120° and 240°, respectively, and three mirror reflections θ1, θ2, θ3 across the three lines passing through the vertices of T and through O (see Fig. 1).
It is edifying to cut a cardboard triangle, label the vertices, and determine composition products explicitly. The result is the "multiplication table" (Fig. 2) for D3.
A routine inspection of the table shows that each element has an inverse, and also (if enough time is spent) that the operation is associative. Associativity is also clear from the fact that each element of D3 is a permutation of the points of the plane. Thus D3 is a group.
If we let S = {1, 2, 3} be the set of vertices of T, then each element of D3 gives rise to a permutation of S, i.e., to an element of the symmetric group S3. For example, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], etc. The result is a 1-1 correspondence between the group D3 of symmetries of T and the symmetric group S3. It is instructive to label the elements of S3 accordingly [e.g., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] etc.], to write out the multiplication table for S3 and to compare with the table above.
7. This time let T be a square in the plane, with center O, and let D4 be its set (in fact group) of symmetries. There are four rotations (one of them the identity, through 0°) and four reflections (see Fig. 3). The multiplication table should be computed.
Again each element of D4 gives rise to a permutation of the set S = {1, 2, 3, 4} of vertices of T, i.e., to an element of S4. For example, the rotation φ1 through 90° counterclockwise about O gives the permutation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Note in this case, however, that not all elements of S4 occur. For example, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not the result of any symmetry of the square.
8. The quaternion group Q2 consists of 8 matrices ± 1, ±i, ±j, ±k under multiplication, where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and 1 denotes the 4 × 4 identity matrix. It is easy to verify that i2 = j2 = k2 = — 1 and that ij = k. All other products can be determined from those. For example, since ijk = k2 = -1 we have i2jk = -jk = -i, and hence jk = i. The chief advantage of presenting Q2 as a set of matrices is that the associative law is automatically satisfied.
9. Klein's 4-group K consists of four 2 × 2 matrices:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Its multiplication table is Fig. 4.
10. Let T be a regular tetrahedron and let G be the set of all rotations of three-dimensional space that carry T to itself (as a set of points), i.e., all the rotational symmetries of T. Thus G consists of the identity 1, rotations through angles of 180° about each of three axes joining midpoints of opposite edges, and rotations through 120° and 240° about each of four axes joining vertices with centers of opposite faces. Thus |G| = 12.
Exercise 1.2. Let G be the set of 12 rotational symmetries of a regular tetrahedron.
(1) Verify that G is a group and write out its multiplication table.
(2) Each element of G gives rise to a permutation of the set of vertices of the tetrahedron, numbered 1, 2, 3, and 4. List the resulting permutations in S4.
(3) Each element of G also gives rise to a permutation of the set of 6 edges of the tetrahedron. List the resulting permutations in S6.
Exercise 1.3. Describe the groups of rotational symmetries of a cube (there are 24) and of a regular dodecahedron (there are 60). It will be helpful to have cardboard models.
Many more examples will appear as we continue. It will be convenient at this point to introduce some concepts, some terminology, and some elementary consequences of the definitions.
The cardinality |G| of a group G is called its order. If G is not finite we usually say simply that G has infinite order. An easy counting argument shows that the symmetric group Sn has order n!.
(Continues...)
Excerpted from ALGEBRA by Larry C. Grove. Copyright © 1983 Larry C. Grove. Excerpted by permission of Dover Publications, Inc..
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