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**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**_{pa} ×
**Z**_{qb} ×
** · · · **
× **Z**_{tk} .

**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.

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
**Z**_{6},
**Z**_{9}^{×}, and
**Z**_{18}^{×} are isomorphic to each other.
*Solution*

**21.**
Is **Z**_{4} × **Z**_{10}
isomorphic to **Z**_{2} × **Z**_{20}?
*Solution*

**22.**
Is **Z**_{4} × **Z**_{15}
isomorphic to **Z**_{6} × **Z**_{10}?
*Solution*

**23.**
Give the lattice diagram
of subgroups of **Z**_{100}.
*Solution*

**24.**
Find all generators
of the cyclic group **Z**_{28}.
*Solution*

**25.**
In **Z**_{30}, find the order
of the subgroup < [18]_{30} >;
find the order of < [24]_{30} >.
*Solution*

**26.**
Prove that if G_{1} and G_{2} are groups of order 7 and 11,
respectively, then the direct product
G_{1} × G_{2} 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
**Z**_{15}^{×} and
**Z**_{21}^{×}
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 S_{4}
generated by (1,2,3,4) and (1,3).
with a^{2} = (1,3)(2,4) and a^{3} = 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*

**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
Z_{3} Z_{4}
in ** Groups15**.

**Lab 4.**
Give the lattice diagram of subgroups of the group of order 12 denoted by
A_{4}
in ** Groups15**.

Forward to §3.6 | Back to §3.4 | Up | Table of Contents

Forward to §3.6 | Back to §3.4 | Up | Table of Contents