Introduction to Groups, Chapter 2

plane symmetry

A plane symmetry of a figure F in a plane is a function from the plane to itself that carries F onto F and preserves distances; that is, for any points p and q in the plane, the distance from the image of p to the image of q is the same as the distance from p to q.

reflection across a line L

A reflection across a line L is that function that leaves every point of L fixed and takes any point q, not on L, to the point q' so that L is the perpendicular bisector of the line segment joining q and q'. Thus, in the dihedral groups, the motions that we described as flips about axes of symmetry in three dimensions (for example, H, V, D, U) are reflections across lines in two dimensions. Many objects and figures have rotational symmetry but not reflective symmetry.

cyclic rotation group of order n

A symmetry group consisting of the rotational symmetries of 0°, 360°/n, 2(360°)/n, . . . , (n - 1)360°/n, and no other symmetries.

Abelian

Another property of D_4 deserves special comment. Observe that HD ≠ DH but R_90 R_180 = R_180R_90. Thus, in a group, ab may or may not be the same as ba. If it happens that ab = ba for all choices of group elements a and b, we say the group is commutative or—better yet— Abelian. Otherwise, we say the group is non-Abelian.

Requirements for a Mathematical System to be a Group

Any sequence of motions turns out to be the same as one of these eight. Algebraically, this says that if A and B are in D_4, then so is AB. This property is called closure. if A is any element of D_4, then AR_0 = R_0A = A. Thus, combining any element A on either side with R_0 yields A back again. An element R_0 with this property is called an identity, and every group must have one. we see that for each element A in D_4, there is exactly one element B in D_4 such that AB=BA=R_0. In this case, B is said to be the inverse of A and vice versa. For example, R_90 and R_270 are inverses of each other, and H is its own inverse. The term inverse is a descriptive one, for if A and B are inverses of each other, then B "undoes" whatever A "does," in the sense that A and B taken together in either order produce R_0, representing no change. every element of D_4 appears exactly once in each row and column. This feature is something that all groups must have, and, indeed, it is quite useful to keep this fact in mind when constructing the table in the first place.

Cayley Table for D_4

Denoted by H*R_90, the function composition f*g means "g followed by f." H*R_90 = D.

The Dihedral Groups

The analysis carried out above for a square can similarly be done for an equilateral triangle or regular pentagon or, indeed, any regular n-gon (n ≥ 3). The corresponding group is denoted by D_n and is called the dihedral group of order 2n. The dihedral group of order 2n is often called the group of symmetries of a regular n-gon.

Symmetries of a Square

The eight motions R_0, R_90, R_180, R_270, H, V, D, U, together with the operation composition, form a mathematical system called the dihedral group of order 8 (the order of a group is the number of elements it contains). Denoted by D_4.

Associativity of a Group

The remaining condition required for a group is associativity; that is, (ab)c = a(bc) for all a, b, c in the set. To be sure that D_4 is indeed a group, we should check this equation for each of the 8³ = 512 possible choices of a, b, and c in D_4. In practice, however, this is rarely done! We simply observe that the eight motions are functions and the operation is function composition. Then, since function composition is associative, we do not have to check the equations.

symmetry group

The symmetry group of a plane figure is the set of all symmetries of the figure.