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# EX1-2010 - Algebra 3 2010 Exercises in group theory...

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Algebra 3 2010 Exercises in group theory February 2010 Exercise 1*: Discuss the Exercises in the sections 1.1-1.3 in Chapter I of the notes. Exercise 2: Show that an infinite group G has to contain a non-trivial subgroup, i.e. a subgroup 6 = G, { e } . Exercise 3: Suppose that a 2 b 2 = ( ab ) 2 for all a, b in the group G. Show that G is abelian. Exercise 4*: Show that h (1 , 2 , 3 , 4) , (1 , 2) i = S 4 . Exercise 5: Let a , b and c be elements in a group G . (1) Show that a and a - 1 have the same order. (2) Show that ab and ba have the same order. (3) Show that abc and bca have the same order. (Try to generalize this to a general result.) (4) Find three elements a, b, c in the symmetric group S 3 , such that abc and bac have different orders. Exercise 6*: (1) Let a and b be commuting elements in G of finite orders m and n, re- spectively. Show that ( ab ) mn = e. (2) Suppose in addition that n and m are relatively prime. Prove that if a power a i of a equals a power b j of b , then a i = b j = e . Use this to prove that the order of ab is nm. Exercise 7: Show that the elements of finite order in an abelian group G form a subgroup of G . Exercise 8: Consider the following matrices as elements in the group GL (2 , R ): 1

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a = 0 1 - 1 - 1 b = 0 - 1 1 0 Show that a has order 3 and that b has order 4. Show that the order of ab is infinite. Exercise 9: Give examples of the following: (1) An infinite group G with a subgroup H 6 = G and | G : H | finite. (2) An infinite group G with a subgroup H 6 = { e } and | H | finite. Exercise 10*: Suppose that the elements a, b in the group G satisfy the relation aba - 1 = b 2 , where b 6 = e. (1) Show that a 5 ba - 5 = b 32 . (2) Assume that | a | = 5 . Compute | b | . Exercise 11: Define an equivalence relation on the elements of the group G by a c b ⇐⇒ h a i = h b i . Here h a i is the cyclic subgroup of G generated by a. Show that the c - equivalence class of an element a is always finite. Exercise 12*: Let G be a finite group. (1) Show that if | G | is even then the number of elements of order 2 in G is odd . (Consider the mapping x 7→ x - 1 . Which elements are the fixed points of this map?) (2) Show that the number of elements of order 3 in G is even (possibly 0).
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