Tutorial IV: Group Theory 1. If G is a group in which (a.b)i = ai .bi for three consecutive integers i for all a, b ∈ G then G must be abelian. 2. In S3 give an example of two elements x, y such that (x.y)2 6= x2 .y 2 . 3. In S3 show that there are four elements statisfying x2 = e and three elements satisfying y 3 = e. 4. If G is a finite group, show that there exists a positive integer N such that aN = e for a ∈ G. 5. (a) If the group G has three elements, show that it must be abelian. (b) Do part (a) if G has four elements. (c) Five elements. !

a b 6. (a) Let G be the group of all 2 × 2 matrices where ad − bc 6= 0 and c d a, b, c, d are integers modulo 3, relative to the matrix multiplication. Show that |G| = 48. (b) If we modify the example of G in part (a) by insisting that ad − bc = 1, then what is |G|? 7. Show that a group with no proper non-trivial subgroups is cyclic. 8. Find the order of the cyclic subgroup of the given group generated by the indicated elements: (a) The subgroup of Z4 generated by 3. + isin 2π where U6 is the multiplicative (b) The subgroup of U6 generated by cos 2π 3 3 group of all the sixth roots of unity. 9. Suppose that H is a subgroup of G such that whenever Ha 6= Hb then aH 6= bH. Prove that gHg −1 ⊂ H for all g ∈ G. 10. If H is a subgroup of G, then by the centralizer C(H) of H we mean the set {x ∈ G|xh = hx all h ∈ H}. Prove that C(H) is a subgroup of G. 11. Suppose φ is a homomorphism from a group G to another group H. Prove that the kernel of φ is a normal subgroup. 12. Suppose G is a group of order 48 and H is a group of order 12. Let φ be an onto homomorphism from G onto H. Find the order of the kernel of φ.

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