Forward to §7.6 | Back to §7.4 | Up | Table of Contents | About this document

**Theorem 7.5.1.**
A finite abelian group can be expressed as a direct product of
its Sylow p-subgroups.

**Lemma 7.5.2.**
Let G be a finite abelian p-group, let
a be an element of maximal order in G,
and let b<a> be any coset of G/<a>.
Then there exists d in G such that d<a> = b<a> and
<a> <d> = {e}.

**Lemma 7.5.3.**
Let G be a finite abelian p-group.
If <a> is a cyclic subgroup of G of maximal order,
then there exists a subgroup H with
G <a> × H.

**Theorem 7.5.4.
[Fundamental Theorem of Finite Abelian Groups]**
Any finite abelian group is isomorphic to a direct product
of cyclic groups of prime power order.
Any two such decompositions have the same number of factors of each order.

**Proposition 7.5.5.**
Let G be a finite abelian group.
Then G is isomorphic to a direct product of cyclic groups

**Z**_{n1}
×
**Z**_{n2}
× ** · · · ** ×
**Z**_{nk}

**Corollary 7.5.6.**
Let G be a finite abelian group.
If a is an element of maximal order in G,
then the order of every element of G is a divisor of the order of a.

**Theorem 7.5.8.**
Let p be an odd prime,
let k be a positive integer,
and let n = p^{k}. Then
**Z**_{n}^{×}
is a cyclic group.

**Theorem 7.5.10.**
If k 3,
and n = 2^{k}, then
**Z**_{n}^{×}
is isomorphic to the direct product
of a cyclic group of order 2 and a cyclic group of order 2^{k-2}.

**Corollary 7.5.11.**
The group **Z**_{n}^{×} is cyclic
if and only if
n is of the form 2, 4, p^{k}, or
2p^{k} for an odd prime p.

In elementary number theory,
an integer g is called a
** primitive root**
for the modulus n if
**Z**_{n}^{×}
is a cyclic group and [g]_{n}
is a generator for
**Z**_{n}^{×}.
Corollary 7.5.11
determines which moduli n have primitive roots.
The proof of Theorem 7.5.8
shows how to find a generator for
**Z**_{n}^{×},
where n = p^{k}.

**7.**
Find all abelian groups of order 108 (up to isomorphism).
*Solution*

**8.**
Let G and H be finite abelian groups,
and assume that G × G
is isomorphic to H × H.
Prove that G is isomorphic to H.
*Solution*

**9.**
Let G be an abelian group which has 8 elements of order 3,
18 elements of order 9, and no other elements besides the identity.
Find (with proof) the decomposition of G
as a direct product of cyclic groups.
*Solution*

**10.**
Let G be a finite abelian group such that |G| = 216.
If | 6 G | = 6, determine G up to isomorphism.
*Solution*

**11.**
Apply both structure theorems
to give the two decompositions of the abelian group
**Z**_{216}^{×}.
*Solution*

**12.**
Let G and H be finite abelian groups,
and assume that they have the following property.
For each positive integer m,
G and H have the same number of elements of order m.
Prove that G and H are isomorphic.
*Solution*

Forward to §7.6 | Back to §7.4 | Up | Table of Contents