hw3sol - Math 113 Homework 3 Solutions 1 Fraleigh section 6 exercise 32(as always carefully justify your answers Mark each of the following true or

hw3sol - Math 113 Homework 3 Solutions 1 Fraleigh section 6...

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Math 113 Homework 3 – Solutions 1. Fraleigh, section 6, exercise 32 (as always, carefully justify your answers): Mark each of the following true or false. a. Every cyclic group is abelian. True: any cyclic group has a generator a , and a m a n = a m + n = a n a m (see also Theorem 6.1 in Fraleigh). b. Every abelian group is cyclic. False: the Klein 4-group is abelian but not cyclic. c. ( Q , +) is a cyclic group. False. Indeed, the subgroup ( a ) generated by an element a Q consists of all integer multiples of a . When a = 0, we get ( a ) = { 0 } ; when a negationslash = 0, the subgroup ( a ) does not contain a 2 Q . So in all cases ( a ) is a proper subgroup of Q , hence Q is not cyclic (it is not generated by one of its elements). d. Every element of every cyclic group generates the group. False, for example 2 Z 4 does not generate Z 4 . Another example: for n 2 the element 0 does not generate Z n . e. There is at least one abelian group of every finite order > 0 . True: for any integer n> 0, ( Z n , +) is an abelian group of order n . f. Every group of order 4 is cyclic. False: the Klein 4-group is not cyclic. g. All generators of Z 20 are prime numbers. False: 9 is a generator of Z 20 , but 9 isn’t prime. Since we have defined Z 20 to be a set of congruence classes, the statement doesn’t even quite make sense anyway, because being prime is a property of an integer, while being a generator of Z 20 is a property of a congruence class. (For example, 7 = 27 Z 20 , however the integer 7 is prime while 27 is not). In general you should be careful to distinguish between integers and their equivalence classes mod n whenever it is not clear from the context. h. If G and G are groups, then G G is a group. False. G and G might not intersect at all (if their elements had completely different names!); and even if they do intersect, G G may not have a well-defined binary operation on it. Namely, the group operations on G and G might not agree on G G : for instance how do you intersect ( R + , · ) with ( Z , +)? So it doesn’t even make sense to ask whether G G is a group. i. If H and K are subgroups of a group G , then H K is a group. True, by section 5 exercise 54 (cf. homework 2). j. Every cyclic group of order > 2 has at least two distinct generators. True. Any finite cyclic group of order n > 2 is isomorphic to Z n , where both 1 and 1 are generators (distinct since we assumed that n> 2; for most n there are other generators too). Any infinite cyclic group is isomorphic to Z , which also has two distinct generators, namely 1 and 1. 2. Fraleigh, section 6, exercises 44 and 56(a). # 44: Let G be a cyclic group with generator a , and let G be a group isomorphic to G . If φ : G G is an isomorphism, show that, for every x G , φ ( x ) is completely determined by the value φ ( a ) . That is, if φ : G G and ψ : G G are two isomorphisms such that φ ( a ) = ψ ( a ) , then φ ( x ) = ψ ( x ) for all x G .
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  • Spring '08
  • OGUS
  • Math, Ri, Zn, Cyclic group, Si Sj

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