School of Mathematics and Statistics
MT4003 Groups
Problem Sheet IV: Cyclic Groups
北美数学作业代写机构 Prove that if X is a fifinite subset of Q then the subgroup 〈X〉 is contained in a cyclic subgroup of Q. Conclude that h〈X〉 is cyclic.
1.For each of the following orders n, describe the subgroup structure of the fifinite cyclic group of order n:
(i) 12; (ii) 30; (iii) p2q2
where p and q are distinct primes.
2.(a) Let G be a cyclic group and H be a subgroup of G. Show that G/H is also cyclic.
(b) Now suppose G is a cyclic group of order n and that H is of order d. What is G/H?
3.Let G = C12, a cyclic group of order 12. 北美数学作业代写机构
(a) For each divisor d of 12, list the number of elements of G of order d.
(b) How many elements x are there such that G = hxi?
(c) How many sets A are there with |A| = 2 and G = hAi?
[Hint: Draw the subgroup diagram for G and consider to which subgroups elements of particular order must belong.]
4.Let G = Cpn , a cyclic group of order pn, where p is prime.
(a) For each i = 0, 1, . . . , n, list how many elements there are in G of order pi.
(b) How many elements x are there such that G = 〈x〉?
(c) How many sets A are there with |A| = 2 and G = 〈A〉?
5.Let G = hxi be an infifinite cyclic group. 北美数学作业代写机构
(a) Let H be a non-trivial subgroup of G. Show that H contains an element of the form xk for some k > 0.
(b) Show that H = hxmi, where m is the least positive integer such that xm 2 H. 北美数学作业代写机构
(c) Show that every coset of H has the form Hxi where i 2 {0, 1,…,m− 1}. Show that |G : H| =m.
(d) Deduce that if m is a positive integer then H = hxmi is the unique subgroup of G of index m.
(e) Show that G/H≅ Zm.
[Hint: For (b), mimic the second half of the corresponding proof for fifinite cyclic groups. Use division, with quotient and remainder, in Z.]
6.Let G be any group and N be a normal subgroup that is cyclic. Show that every subgroup of N is also normal in G. 北美数学作业代写机构
Is it true that if G is any group and N is an abelian normal subgroup of G, then every subgroup of N is also normal? If not, provide a counterexample.
7.Consider the group Q of rational numbers, under addition.
(a) Prove that if X is a fifinite subset of Q then the subgroup 〈X〉 is contained in a cyclic subgroup of Q. Conclude that h〈X〉 is cyclic.
(b) Is Q cyclic? Conclude that Q cannot be generated by a fifinite set.
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