WebMath 122 - Problem Set 2 Due Wednesday, Sept 18 1. Recall from class that D 2n is the dihedral group, with presentation D 2n = hr;sjrn = s2 = 1;rs= sr 1i (a) If n= 2kis even and n 4, show that rk is the only nonidentity element of D 2n that commutes with all elements of D 2n. (b) If nis odd and n 3, show that the identity is the only element ... WebMath 122, Solution Set No. 3 1 3.1 Problem 1 (a) This is a subspace. The zero matrix is symmetric, and the set is closed under addition and scalar multiplication because for …
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WebMath 122, Solution Set No. 5 1 5.2.13 (a) (⇒) if x is on l, then glide reflection acts on points on l as a translation; therefore, x,m(x) and m2(x) lie on l and are colinear. (⇐) If x is not on l, then the line joining x to m(x) crosses l (because m is a glide reflection) and so does the line from m(x) to m2(x). If these points are ... WebMath 122 Notes 6 Example 1.4. Here’s another example based on a group we’ve seen before. Let G= S 3 be the symmetry group on 3 letters. Recall that S 3 = fbijections f1;2;3g!f1;2;3g: Set ˙;˝2S 3 to be the bijections ˙= 0 @ 1 7!1 2 7!3 3 7!2 1 A; ˝= 0 @ 1 7!2 2 7!1 3 7!3 1 A: To check that ˙˝= ˙ ˝6= ˝ ˙= ˝˙, let’s check what ... glut2 is an example of a n
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Web(b) Let G = GL 2(R) and consider the matrices A = 0 2 1 2 0 and B = 0 3 3 0Then we have A2 = B2 = I, that is, A and B have order 2, which is finite. However, AB = 2/3 0 0 3/2 which has the startling property that (AB)n = (2/3)n 0 0 (3/2)n which is never equal to the identity for n 6= 0, i.e. AB has infinite order. ♦ WebMATH 122 at Harrisburg Area Community College, Lancaster (HACC Lancaster) in Lancaster, Pennsylvania. Continues the topics covered in MATH 121 pertaining to … WebMath 122, Solution Set No. 2 As a general note, all elements of Sn will be written in disjoint cycle notation unless otherwise specified. Also, as a notational convention, H • G means H is a subgroup of G. 1 2.3.14 (a) Note that, if ’: Z+! Z+ is an automorphism, ’ is completely determined by ’(1). glut4 ea.hy926