Get Cyclic Group Pdf

Math 343 Intro to Algebraic Structures Spring 2010 Homework 7 Solutions p. 110 14 a The cyclic subgroup of Z24 generated by 18 has order 4. b Z3 Z4 is of order 12. c The element 4 2 of Z12 Z8 has order 12. d The Klein 4-group is isomorphic to Z2 Z2. e Z2 Z Z4 has eight elements of nite order. p* 112 39 Let G be an abelian group* Show that the elements of nite order in G form a subgroup* This subgroup is called the torsion subgroup of G* Proof* Let T g G g has nite order. We will prove T is a subgroup of G* We rst note that the identity element e of G has nite order the order of e is 1. So e T. Now suppose a b T. This means there exist positive integers m and n such that am e and bn e. We will prove ab T. Since G is abelian we have ab mn amn bmn am n bn m e n e m e. The last equation implies ab has nite order that is ab T. Finally we show a 1 T. Consider a 1 m a m am 1 e 1 We conclude that T is a subgroup of G* p* 133 6 Let R R where R is additive and R is multiplicative be given by x 2x. Determine if is a homomorphism* Claim is a homomorphism* Proof* Let x y R* Then x y 2x y 2x 2y x y. p* 133 12 Let Mn be the additive group of all n n matrices with real entries and let R be the additive group of real numbers. Let A det A the determinant of A for A Mn. Determine if is a homomorphism* Claim is not a homomorphism* Proof* In general we know that det A B is not equal to det A det B for n n matrices A and B. For an example suppose n 2 and let I denote the 2 2 identity matrix. Then det I I det 0 0 but det I det I 1 1 2. p* 135 47 Show that any group homomorphism G G where G is a prime must either be the trivial homomorphism or a one-to-one map* Proof* Let G G be a group homomorphism and assume that G p where p is a prime number. Since Ker is a subgroup of G we know the order of Ker divides p* Since p is prime the order of Ker is either 1 or p* If the order of Ker is 1 then Ker e where e is the identity element of G* Now Corollary 13. 18 shows is one-to-one. trivial homomorphism since every element of G is sent to the identity e of G*. b Z3 Z4 is of order 12. c The element 4 2 of Z12 Z8 has order 12. d The Klein 4-group is isomorphic to Z2 Z2. e Z2 Z Z4 has eight elements of nite order. p* 112 39 Let G be an abelian group* Show that the elements of nite order in G form a subgroup* This subgroup is called the torsion subgroup of G* Proof* Let T g G g has nite order. e Z2 Z Z4 has eight elements of nite order. p* 112 39 Let G be an abelian group* Show that the elements of nite order in G form a subgroup* This subgroup is called the torsion subgroup of G* Proof* Let T g G g has nite order. We will prove T is a subgroup of G* We rst note that the identity element e of G has nite order the order of e is 1. We will prove T is a subgroup of G* We rst note that the identity element e of G has nite order the order of e is 1. So e T. Now suppose a b T. This means there exist positive integers m and n such that am e and bn e. So e T. Now suppose a b T. This means there exist positive integers m and n such that am e and bn e. We will prove ab T. Since G is abelian we have ab mn amn bmn am n bn m e n e m e. The last equation implies ab has nite order that is ab T.