Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

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§ 3.5 Cyclic Groups

 
Theorem 3.5.1. Every subgroup of a cyclic group is cyclic.

 
Theorem 3.5.2. Let G cyclic group.

(a) If G is infinite, then G Z.

(b) If |G| = n, then G Zn.

 
Proposition 3.5.3. Let G = <a> be a cyclic group with |G| = n.

(a) If m is any integer, then <am> = <ad>, where d=gcd(m,n), and am has order n/d.

(b) The element ak generates G if and only if gcd(k,n)=1.

(c) The subgroups of G are in one-to-one correspondence with the positive divisors of n.

(d) If m and k are divisors of n, then <am> <ak> if and only if k | m.

 
Theorem 3.5.4. If n = pa qb · · · tk is the prime decomposition of the positive integer n, where p < q < . . . < t are prime numbers, then

Zn Zpa × Zqb × · · · × Ztk .

 
Corollary 3.5.5. If n = pa qb · · · tk is the prime decomposition of the positive integer n, where p < q < . . . < t are prime numbers, then

(n) = n (1 - 1/p) (1 - 1/q) · · · (1 - 1/t) .

 
Definition 3.5.6. Let G be a group. If there exists a positive integer N such that aN=e for all a in G, then the smallest such positive integer is called the exponent of G.

 
Lemma 3.5.7. Let G be a group, and let a,b be elements of G such that ab = ba. If the orders of a and b are relatively prime, then o(ab) = o(a)o(b).

 
Proposition 3.5.8. Let G be a finite abelian group.

(a) The exponent of G is equal to the order of any element of G of maximal order.

(b) The group G is cyclic if and only if its exponent is equal to its order.



§ 3.5 Cyclic Groups: Solved problems

We began our study of abstract algebra very concretely, by looking at the group Z of integers, and the related groups Zn. We discovered that each of these groups is generated by a single element, and this motivated the definition of an abstract cyclic group. In this section, Theorem 3.5.2 shows that every cyclic group is isomorphic to one of these concrete examples, so all of the information about cyclic groups is already contained in these basic examples. One of the useful consequences is that two finite cyclic groups are isomorphic if and only if they have the same number of elements.

You should pay particular attention to Proposition 3.5.3, which describes the subgroups of Zn, showing that they are in one-to-one correspondence with the positive divisors of n. In n is a prime power, then the subgroups are "linearly ordered" in the sense that given any two subgroups, one is a subset of the other. These cyclic groups have a particularly simple structure, and form the basic building blocks for all finite abelian groups. (In Theorem 7.5.4 we will prove that every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order.)


20. Show that the three groups Z6, Z9×, and Z18× are isomorphic to each other.     Solution

21. Is Z4 × Z10 isomorphic to Z2 × Z20?     Solution

22. Is Z4 × Z15 isomorphic to Z6 × Z10?     Solution

23. Give the lattice diagram of subgroups of Z100.     Solution

24. Find all generators of the cyclic group Z28.     Solution

25. In Z30, find the order of the subgroup < [18]30 >; find the order of < [24]30 >.     Solution

26. Prove that if G1 and G2 are groups of order 7 and 11, respectively, then the direct product G1 × G2 is a cyclic group.     Solution

27. Show that any cyclic group of even order has exactly one element of order 2.     Solution

28. Use the the result in Problem 27 to show that the multiplicative groups Z15× and Z21× are not cyclic groups.     Solution

29. Find all cyclic subgroups of the quaternion group. Use this information to show that the quaternion group cannot be isomorphic to the subgroup of S4 generated by (1,2,3,4) and (1,3). with a2 = (1,3)(2,4) and a3 = a-1 = (1,4,3,2).     Solution

30. Prove that if p and q are different odd primes, then Zpq× is not a cyclic group.     Solution


§ 3.5 Lab questions

To answer the experimental questions, use the Groups15 applet written by John Wavrik of UCSD.

Lab 1. (a) Find the exponent of each group listed in Groups15.
(b) List the nonabelian groups that provide a counterexample to Proposition 3.5.8 (a).
(c) List the nonabelian groups that provide a counterexample to Proposition 3.5.8 (b).

Lab 2. Give the lattice diagram of subgroups of the group of order 8 denoted by Q in Groups15.

Lab 3. Give the lattice diagram of subgroups of the group of order 12 denoted by Z3 Z4 in Groups15.

Lab 4. Give the lattice diagram of subgroups of the group of order 12 denoted by A4 in Groups15.


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