Group Axioms, Subgroups and Cyclic Structure

A binary operation on a set G is a function ∗: G × G →G. In more detail, a binary operation assigns an element ∗(x, y) in G to each ordered pair (x, y) of elements in G. It is more natural to write x ∗y instead of ∗(x, y); thus, composition of functions is the function (g, f ) ↦g ◦f ; multiplication, addition, and subtraction are, respectively, the functions (x, y) ↦xy, (x, y) ↦x + y, and (x, y) ↦ x −y. The examples of composition and subtraction show why we want ordered pairs, for x ∗y and y ∗x may be distinct. As with any function, a binary operation is well-defined; when one says this explicitly, it is usually called the law of substitution: If x = x′ and y = y′, then x ∗y = x′ ∗y′.

A group is a set G equipped with a binary operation ∗such that (i) the associative law holds: for every x, y, z ∈G, x ∗(y ∗z) = (x ∗y) ∗z; (ii) there is an element e ∈G, called the identity, with e ∗x = x = x ∗e for all x ∈G; (iii) every x ∈G has an inverse; there is x′ ∈G with x ∗x′ = e = x′ ∗x. By Theorem 1.49, the set SX of all permutations of a set X, with composition as the operation and 1X = (1) as the identity, is a group (the symmetric group on X). This is a useful observation, for it is more efficient, when verifying that a set with an operation is actually a group, to check fewer equations. We are now at the precise point when algebra becomes abstract algebra. In contrast to the concrete group Sn consisting of all the permutations of {1, 2, . . . , n}, we have passed to groups whose elements are unspecified. Moreover, products of elements are not explicitly computable but are, instead, merely subject to certain rules. It will be seen that this approach is quite fruitful, for theorems now apply to many different groups, and it is more efficient to prove theorems once for all instead of proving them anew for each group encountered. In addition to this obvious economy, it is often simpler to work with the “abstract” viewpoint even when dealing with a particular concrete group.

A group G is called abelian3 if it satisfies the commutative law: x ∗y = y ∗x holds for every x, y ∈G. The groups Sn, for n ≥3, are not abelian because (1 2) and (1 3) are elements of Sn that do not commute: (1 2)(1 3) = (1 3 2) and (1 3)(1 2) = (1 2 3).

An expression a1a2 · · · an needs no parentheses if all the ultimate products it yields are equal; that is, no matter what choices are made of adjacent factors to multiply, all the resulting products in G are equal.

Let G be a group and let a ∈G. If ak = 1 for some k ≥1, then the smallest such exponent k ≥1 is called the order of a; if no such power exists, then one says that a has infinite order. The additive group of integers, Z, is a group, and 3 is an element in it having infinite order (because 3 + 3 + · · · + 3 is never 0). In any group G, the identity has order 1, and it is the only element of order 1; an element has order 2 if and only if it is equal to its own inverse. The definition of order says that if x has order n and xm = 1 for some positive integer m, then n ≤m. The next theorem says that n must be a divisor of m.

A motion is a distance preserving bijection ϕ : R2 →R2 [it can be shown that ϕ is a linear transformation if ϕ(0) = 0]. If π is a polygon in the plane, then its symmetry group (π) consists of all the motions ϕ for which ϕ(π) = π. The elements of (π) are called symmetries of π.

If πn is a regular polygon with n vertices v1, v2, . . . , vn and center O, then the symmetry group (πn) is called the dihedral group with 2n elements, and it is denoted6 by D2n. The dihedral group D2n contains the n rotations ρ j about the center by (360 j/n)◦, where 0 ≤j ≤n −1. The description of the other n elements depends on the parity of n. . . , n. . . , q, where mi is the midpoint of the edge vivi+1. For example, the six lines of symmetry of π6 are Ov1, Ov2, and Ov3, and Om1, Om2, and Om3.

A subset H of a group G is a subgroup if (i) 1 ∈H; (ii) if x, y ∈H, then xy ∈H; (iii) if x ∈H, then x−1 ∈H. If H is a subgroup of G, we write H ≤G; if H is a proper subgroup of G, that is, H ̸= G, then we write H < G.

Let G be a group. (i) The cancellation laws hold: If either x ∗a = x ∗b or a ∗x = b ∗x, then a = b. (ii) The element e is the unique element in G with e ∗x = x = x ∗e for all x ∈G. (iii) Each x ∈G has a unique inverse: There is only one element x′ ∈G with x ∗x′ = e = x′ ∗x (henceforth, this element will be denoted by x−1). (iv) (x−1)−1 = x for all x ∈G.

If G is a group and a1, a2, . . . , an ∈G, then the expression a1a2 · · · an needs no parentheses. Remark. This result holds in greater generality, for neither the identity element nor inverses will be used in the proof. ◀

Let G be a group, let a, b ∈G, and let m and n be (not necessarily positive) integers. (i) If a and b commute, then (ab)n = anbn. (ii) (an)m = amn. (iii) aman = am+n.

Let α ∈Sn. (i) If α is an r-cycle, then α has order r. (ii) If α = β1 · · · βt is a product of disjoint ri-cycles βi, then α has order lcm{r1, . . . ,rt}. (iii) If p is a prime, then α has order p if and only if it is a p-cycle or a product of disjoint p-cycles.

If G is a finite group, then every x ∈G has finite order. Table 2.3 augments the table in Example 2.5(ii). Cycle Structure Number Order Parity (1) Even (1 2) Odd (1 2 3) Even (1 2 3 4) Odd (1 2 3 4 5) Even (1 2)(3 4 5) Odd (1 2)(3 4) Even Table 2.3. Permutations in S5 Here are some geometric examples of groups.

(i) The set Q× of all nonzero rationals is an abelian group, where ∗is ordinary multiplication, the number 1 is the identity, and the inverse of r ∈Q× is 1/r. Similarly, R× and C× are multiplicative abelian groups. Note that the set Z× of all nonzero integers is not a multiplicative group, for none of its elements (aside from ±1) has a multiplicative inverse which is an integer. (ii) The set Z of all integers is an additive abelian group with a ∗b = a + b, with identity e = 0, and with the inverse of an integer n being −n. (iii) The circle group, S1 = {z ∈C : |z| = 1}, is the group whose operation is multiplication of complex numbers; this is an operation because the product of complex numbers of modulus 1 also has modulus 1, by Corollary 1.31. Complex multiplication is associative, the identity is 1 (which has modulus 1), and the inverse of any complex number of modulus 1 is its complex conjugate, which also has modulus 1. Therefore, S1 is a group. (iv) For any positive integer n, let µn = ζ k : 0 ≤k < n be the set of all the nth roots of unity, where ζ = e2πi/n = cos ( 2π n ) + i sin ( 2π n ) . (v) The plane R × R is a group with operation vector addition; that is, if α = (x, y) and α′ = (x′, y′), then α + α′ = (x + x′, y + y′). The identity is the origin O = (0, 0), and the inverse of (x, y) is (−x, −y). ◀

(i) Let π4 be a square having sides of length 1 and vertices {v1, v2, v3, v4}; draw π4 in the plane so that its center is at the origin O and its sides are parallel to the axes. It can be shown that every ϕ ∈ (π4) permutes the vertices; indeed, a symmetry ϕ of π4 is determined by {ϕ(vi) : 1 ≤i ≤4}, and so there are at most 24 = 4! possible symmetries. Not every permutation in S4 arises from a symmetry of π4, however. If vi and v j are adjacent, then ∥vi −v j∥= 1, but ∥v1 −v3∥= √ 2 = ∥v2 −v4∥; it follows that ϕ must preserve adjacency (for motions preserve distance). The reader may now check that there are only eight symmetries of π4. Aside from the identity and the three rotations about O by 90◦, 180◦, and 270◦, there are four reflections, respectively, in the lines v1v3, v2v4, the x-axis, and the y-axis (for a generalization to come, note that the y-axis is Om1, where m1 is the midpoint of v1v2, and the x-axis is Om2, where m2 is the midpoint of v2v3). The group (π4) is called the dihedral group5 with 8 elements, and it is denoted by D8. m v v v v m O Figure 2.4 m v v v v m O v5 m 3 m 4 m 5 Figure 2.5 (ii) The symmetry group (π5) of a regular pentagon π5 with vertices v1, . . . , v5 and center O has 10 elements: the rotations about the origin of (72 j)◦, where 0 ≤j ≤4, as well as the reflections in the lines Ovk for 1 ≤k ≤5. The symmetry group (π5) is called the dihedral group with 10 elements, and it is denoted by D10. ◀ m v v v v m O v5 m 3 v6 Figure 2.6 5F. Some of these occur as symmetry groups of regular polyhedra (from the Greek poly meaning “many” and hedron meaning “two-dimensional side”). He invented a degenerate polyhedron that he called a dihedron, from the Greek words di meaning “two” and hedron, which consists of two congruent regular polygons of zero thickness pasted together. The symmetry group of a dihedron is thus called a dihedral group. For our purposes, it is more natural to describe these groups as in the text. Groups