(2) Short answer questions. Write each answer clearly in the box provided, and note that the answer questions is just a YES or a NO. You need only give the answer and no reasons need be given. (a) If G = D x Z/3Z, and g = (Roo, 1), then what is an element of G, other than g, that is conjugate to g? (b) What is the size of the conjugacy class of (1 2)(3 4 5 6) in So? (c) If the list of the sizes of conjugacy classes of a group G is 1, 1, 2, 2, 3, 3, then what are the possibilities for |Z(G), the size of the center of G? (d) When S₁ acts on S4 by conjugation, then what is the number of elements in the orbit of (1 2)? 4 (e) When S4 acts by conjugation on the set of subgroups of S4, then what is the number of subgroups in the orbit of H = {e, (1 2)}? (f) When S₁ acts on S4 by left multiplication, then what is the number of elements in the orbit of (1 2) (g) If H is a subgroup of a group G and x E G, then is xHx¹ necessarily a subgroup of G isomorphic to H? (h) If H and K are subgroups of G and |G: H = 7 and |G: K| = 5, then is HK = KH necessarily?

(2) Short answer questions. Write each answer clearly in the box provided, and note that the answer questions is just a YES or a NO. You need only give the answer and no reasons need be given. (a) If G = D x Z/3Z, and g = (Roo, 1), then what is an element of G, other than g, that is conjugate to g? (b) What is the size of the conjugacy class of (1 2)(3 4 5 6) in So? (c) If the list of the sizes of conjugacy classes of a group G is 1, 1, 2, 2, 3, 3, then what are the possibilities for |Z(G), the size of the center of G? (d) When S₁ acts on S4 by conjugation, then what is the number of elements in the orbit of (1 2)? 4 (e) When S4 acts by conjugation on the set of subgroups of S4, then what is the number of subgroups in the orbit of H = {e, (1 2)}? (f) When S₁ acts on S4 by left multiplication, then what is the number of elements in the orbit of (1 2) (g) If H is a subgroup of a group G and x E G, then is xHx¹ necessarily a subgroup of G isomorphic to H? (h) If H and K are subgroups of G and |G: H = 7 and |G: K| = 5, then is HK = KH necessarily?

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.6: The Algebra Of Matrices

Problem 38E

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