Fundamental theorem of finite Abelian groups
Every finite Abelian group is a direct product of cyclic
groups of prime-power order. Moreover, the number of
terms in the product and the orders of the cyclic groups
are uniquely determined by the group.
Greedy algorithm for an Abelian group of order 𝑝𝑛
1. Compute the orders of the elements of the group
G.
2. Select an element 𝑎1 of maximum order and
define 𝑮1 = ⟨𝑎1 ⟩. Set i = 1.
3. If |𝑮| = |𝑮𝑖 |, stop. Otherwise, replace i by i+1.
4. Select an element 𝑎𝑖 of maximum order 𝑝𝑘 such
𝑝
𝑝2
that 𝑝𝑘 ≤ |𝑮|/|𝑮𝑖 | and none of 𝑎𝑖 , 𝑎𝑖 , 𝑎𝑖 , …,
𝑝𝑘−1
𝑎𝑖
is in 𝑮𝑖−1 , and define 𝑮𝑖 = 𝑮𝑖−1 × ⟨𝑎𝑖 ⟩.
5. Return to step 3.
Existence of subgroups of Abelian groups
If m divides the order of a finite Abelian group G, then G
has a subgroup of order m.
Lemma 1, p. 216
Let G be a finite Abelian group of order 𝑝𝑛 𝑚, where p is
a prime that does not divide m. Then 𝑮 = 𝑯 × 𝑲, where
𝑛
𝑯 = {𝑥 ∈ 𝑮 | 𝑥 𝑝 = 𝑒} and 𝑲 = {𝑥 ∈ 𝑮 | 𝑥 𝑚 = 𝑒}.
Moreover, |𝑯| = 𝑝𝑛 .
Lemma 2, p. 217
Let G be an abelian group of prime-power order and let
a be an element of maximal order in G. Then G can be
written in the form ⟨𝑎⟩ × 𝑲.
Lemma 3, p. 218
A finite Abelian group of prime-power order is an
internal direct product of cyclic groups.
Lemma 4, p. 218
Suppose that G is a finite Abelian group of prime-power
order. If 𝑮 = 𝑯1 × 𝑯2 × … × 𝑯𝑚 and 𝑮 = 𝑲1 × 𝑲2 × … ×
𝑲𝑛 , where the H’s and K’s are nontrivial cyclic
subgroups with |𝑯1 | ≥ |𝑯2 | ≥ ⋯ ≥ |𝑯𝑚 | and |𝑲1 | ≥
|𝑲2 | ≥ ⋯ ≥ |𝑲𝑛 |, then m=n and |𝑯𝑖 | = |𝑲𝑖 | for all i.