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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 Z_n.

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

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

(b) The element a^k 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 <a^m> <a^k> if and only if k | m.

 
Theorem 3.5.4. If n = p^a q^b · · · t^k is the prime decomposition of the positive integer n, where p < q < . . . < t are prime numbers, then

Z_n Z_p^a × Z_q^b × · · · × Z_t^k .

 
Corollary 3.5.5. If n = p^a q^b · · · t^k 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 a^N=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

You should pay particular attention to Proposition 3.5.3, which describes the subgroups of Z_n, 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 Z_pq^× is not a cyclic group.     Solution

§ 3.5 Lab questions

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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