Section 1.3. Groups 25 Definition. If G is a finite group, then the number of elements in G, denoted by |G|, is called the order of G. The word order in Group Theory has two meanings: the order of an element a ∈ G the order |G| of a group G. Proposition 1.38 will explain this by relating the order of a group element a with the order of a group determined by it. Here are some geometric examples of groups. Definition. An isometry is a distance preserving bijection14 ϕ: R2 → R2 that is, if v − u is the distance from v to u, then ϕ(v) − ϕ(u) = v − u . If π is a polygon in the plane, then its symmetry group Σ(π) consists of all the isometries ϕ for which ϕ(π) = π. The elements of Σ(π) are called symmetries of π. Example 1.30. Let π4 be a square having vertices {v1,v2,v3,v4} and sides of length 1 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 ∈ Σ(π4) permutes the m v1 v3 v2 v4 m 1 2 O Figure 1.5. Square. vertices (Exercise 1.65 on page 46) 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 vj are adjacent, then vi −vj = 1, but v1 −v3 = √ 2 = v2 −v4 it follows that ϕ must preserve adjacency (for isometries 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 Σ(π4) is called the dihedral group15 of order 8, and it is denoted by D8. 14It can be shown that ϕ is a linear transformation if ϕ(0) = 0 (FCAA, Proposition 2.59). A distance preserving function f : R2 → R2 is easily seen to be an injection. It is not so obvious (though it is true) that f must also be a surjection (FCAA, Corollary 2.60). 15Klein was investigating those finite groups occurring as subgroups of the group of isometries of R3. 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 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. It is more natural for us to describe these groups as in the text.

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