Primary Decomposition Theorem for Finite Abelian Groups

Chapter 1 : Direct Product of Groups

Module 3 : Primary decomposition theorem for ﬁnite abelian groups

Anjan Kumar Bhuniya Department of Mathematics Visva-Bharati; Santiniketan West Bengal

1 Primary decomposition theorem for ﬁnite abelian groups

Learning outcomes: 1. p-primary abelian groups. 2. Primary Decomposition Theorem. 3. p-primary components.

In this module we prove that every finite abelian group can be expressed as an internal direct product of a family of p-primary subgroups. In most of the results on the structure of finite abelian groups the feature that plays the most powerful role is that every finite abelian group is a Z-module. So, it is convenient to consider addition as the binary operation in the study of finite abelian groups.

Internal direct product of a family of subgroups G1,G2, ··· ,Gk of a group G, if it is abelian, is called direct sum of G1,G2, ··· ,Gk and is denoted by G1 ⊕ G2 ⊕ · · · ⊕ Gk. Thus a ﬁnite abelian group G is a direct sum of subgroups G1,G2, ··· ,Gk if and only if G = G1 + G2 + ··· + Gk and

Gi ∩(G1 +···+Gi−1 +Gi+1 +···+Gk) = {0} for every i = 1, 2, ··· , k. In view of the fact that every internal direct sum of the subgroups G1,G2, ··· ,Gk is isomorphic to the external direct product of G1,G2, ··· ,Gk, we call both the internal direct sum and the external direct product simply as direct sum and use the notation G1 ⊕G2 ⊕· · ·⊕Gk to represent direct sum of groups G1,G2, ··· ,Gk.

Deﬁnition 0.1. Let p be a prime. An abelian group G is called p-primary if every element of G is of order pm for some m ≥ 0, equivalently, if for each a ∈ G there is m ≥ 0 such that pma = 0.

Lagrange’s Theorem implies that every ﬁnite abelian group G of order pn is a p-primary group. Following result shows that order of every p-primary abelian group is of this form.

Lemma 0.2. Every ﬁnite p-primary abelian group is of order pn for some n ≥ 0.

Proof. Let G = {g1, g2, ··· , gk} be a p-primary abelian group. Then, for any gi ∈ G, gi ∈< gi >⊆< g1 > + < g2 > + ··· + < gk > implies that G ⊆< g1 > + < g2 > + ··· + < gk >. Also,

< g1 > + < g2 > + ··· + < gk >⊆ G. Therefore G =< g1 > + < g2 > + ··· + < gk >. Now we show that |G| = pn by the induction on k, the number of elements of G. If k = 1, then

n G =< g1 > implies that |G| = | < g1 > | = o(g1) = p . Suppose that G has k + 1 elements. Then

2 G =< g1 > + < g2 > + ··· + < gk+1 >, and we have | < g > + < g > + ··· + < g > || < g > | |G| = 1 2 k k+1 |(< g1 > + < g2 > + ··· + < gk >)∩ < gk+1 > | which implies that |G| divides | < g1 > + < g2 > + ··· + < gk > || < gk+1 > |. Since | < g1 > r s + < g2 > + ··· + < gk > | = p , by induction hypothesis, and | < gk+1 > | = o(gk+1) = p , so |G| divides pr+s. Thus it follows that |G| = pn for some n ≥ 0.

Thus the Klein’s 4-group K4 is a 2-primary group. For a prime number p, the group Zp ⊕Zp2 ⊕

Zp3 is a p-primary group. But Z2 ⊕ Z3 is not a p-primary group. Let G be a ﬁnite abelian group and p a prime number. Denote

m Gp = {a ∈ G | p a = 0 for some integer m ≥ 0}.

r Then Gp is the set of all elements of order p for some r ≥ 0 in G. For every prime number p, Gp is a subgroup of G.

Example 0.3. Consider G = Z36. Then

G2 = {[a] ∈ Z36 | o([a]) = 1, 2, or 4} 36 = {[a] ∈ | = 1, 2, or 4} Z36 gcd(a, 36)

= {[a] ∈ Z36 | gcd(a, 36) = 36, 18, or 9} = {[0], [9], [18], [27]}

' Z4.

Similarly, G3 = {[0], [4], [8], [12], [16], [20], [24], [28], [32]} which is isomorphic to Z9.

Also we have G = G2 ⊕ G3.

Theorem 0.4 (Primary Decomposition Theorem). 1. Let G be a ﬁnite abelian group and |G| =

n1 n2 nk p1 p2 ··· pk . Then

(a) G = Gp1 ⊕ Gp2 ⊕ · · · ⊕ Gpk ,

ni (b) |Gpi | = pi for every i.

2. Let G and H be two ﬁnite abelian groups of the same order. Then G ' H if and only if

Gp ' Hp for every prime p | |G|.

3 n1 n2 nk m1 m2 mk Proof. 1. (a) Let x ∈ G. Then o(x) | p1 p2 ··· pk implies that o(x) = p1 p2 ··· pk where

m1 m2 mk d mi 0 ≤ mi ≤ ni. Denote d = p1 p2 ··· pk and ri = mi for every 1 ≤ i ≤ k. Then pi (rix) = pi dx = 0 implies that rix ∈ Gpi for every i. Since gcd(r1, r2, ··· , rk) = 1, so there are si ∈ Z such that 1 = s1r1 + s2r2 + ··· + skrk which implies that x = s1r1x + s2r2x + ··· + skrkx ∈

Gp1 + Gp2 + ··· + Gpk . Therefore G = Gp1 + Gp2 + ··· + Gpk . To show that this sum is a direct

li sum, consider x ∈ Gpi ∩ (Gp1 + Gp2 + ··· Gpi−1 + Gpi+1 + ··· + Gpk ). Then pi x = 0 for some li ≥ 0 and x = x1 + x2 + ··· + xi−1 + xi+1 + ··· + xk where xi ∈ Gpi . Then there is some lj ≥ 0 such that

lj l1 l2 li−1 li+1 lk pj xj = 0 for all j = 1, 2, ··· , i−1, i+1, ··· , k, and hence ux = 0 where u = p1 p2 ··· pi−1 pi+1 ··· pk .

li li li Since gcd(u, pi ) = 1, so there exist r, s ∈ Z such that 1 = ru + spi , and so x = rux + spi x = 0.

Therefore G = Gp1 ⊕ Gp2 ⊕ · · · ⊕ Gpk .

mi (b) Every Gpi is a pi-primary abelian group and hence |Gpi | = pi for some mi ≥ 0. Then

m1 m2 mk n1 n2 nk m1 m2 mk G = Gp1 ⊕Gp2 ⊕· · ·⊕Gpk implies that |G| = p1 p2 ··· pk , that is, p1 p2 ··· pk = p1 p2 ··· pk .

It follows from the Fundamental Theorem of Arithmetic that ni = mi for every i = 1, 2, ··· , k. 2. Suppose that ψ : G −→ H is an isomorphism. Let p be a prime divisor of |G|. Then, for

m m m every a ∈ Gp, p a = 0 implies that p ψ(a) = ψ(p a) = ψ(0) = 0, and so ψ(Gp) ⊆ Hp. Thus

ψ|Gp : Gp −→ Hp is an one-to-one homomorphism. Since |G| = |H|, so |Gp| = |HP |, by 1.(b). Hence

ψ|Gp : Gp −→ Hp is onto. Therefore Gp ' Hp. Converse follows trivially.

Thus for every prime number p | |G|, the subgroups Gp are primary building blocks of the group G.

Deﬁnition 0.5. Let G be a ﬁnite abelian group. Then for every prime divisor p of |G|, Gp is called the p-primary component of G.

Example 0.6. Consider G = Z2 ⊕ Z23 ⊕ Z52 . Then the 2-primary component of G is Z2 ⊕ Z23 and the 5-primary component of G is Z52 .

Example 0.7. Let G be a p-primary ﬁnite abelian group. We show that every nonidentity element of G is of order p if and only if G ' Zp ⊕ Zp ⊕ · · · ⊕ Zp. First assume that every nonidentity element of G is of order p. Since G is p-primary, so it follows that |G| = pn for some n ≥ 1. We prove the result by the principle of mathematical

4 n+1 induction on n. If n = 1, then |G| = p which implies that G ' Zp. Let |G| = p , n ≥ 1. Consider a nonidentity element a of G and denote H =< a >. Then |H| = o(a) = p which implies that

H ' Zp. Now consider

S = {K ⊆ G | K is a subgroup of G such that H ∩ K = {0}}.

Then {0} is in S. Therefore S is nonempty. Also S is a partially ordered set with respect to set inclusion. Since S has only ﬁnite number of elements, so S has a maximal element, say K. Now we show that G = H ⊕ K. Otherwise, H ∩ K = {0} implies that G 6= H + K. Consider c ∈ G \ (H + K). Then c 6∈ K implies that K is properly contained in K+ < c >. Also if x ∈ H ∩ (K+ < c >) then x = h = k + nc for some h ∈ H, k ∈ K and n ∈ N, which implies that h − k = nc ∈< c > ∩(H + K). Now o(c) = p implies that | < c > ∩(H + K)| = 1 or p. If | < c > ∩(H +K)| = p, then < c > ∩(H +K) =< c > and so < c >⊆ H +K which contradicts that c 6∈ H + K. Hence | < c > ∩(H + K)| = 1 and so < c > ∩(H + K) = {0}. Thus h − k = 0 which implies that x = h = k ∈ H ∩ K = {0}. Thus H ∩ (K+ < c >) = {0}, and hence K+ < c >∈ S which contradicts the maximality of K in S. Thus G = H ⊕ K. Then |K| = pn which implies that K ' Zp ⊕ Zp ⊕ · · · ⊕ Zp (n-times), by induction hypothesis. Thus G ' Zp ⊕ Zp ⊕ · · · ⊕ Zp (n + 1-times). Converse follows directly.

1 Summary

• Let p be a prime. An abelian group G is called p-primary if every element of G is of order pm for some m ≥ 0, equivalently, if for each a ∈ G there is m ≥ 0 such that pma = 0.

• Every ﬁnite p-primary abelian group is of order pn for some n ≥ 0.

n1 n2 nk • 1. Let G be a ﬁnite abelian group and |G| = p1 p2 ··· pk . Then

(a) G = Gp1 ⊕ Gp2 ⊕ · · · ⊕ Gpk ,

ni (b) |Gpi | = pi for every i.

2. Let G and H be two ﬁnite abelian groups of the same order. Then G ' H if and only if

Gp ' Hp for every prime p | |G|.

5 This result is known as primary decomposition theorem.

• Let G be a ﬁnite abelian group. Then for every prime divisor p of |G|, Gp is called the p-primary component of G.

• Let G be a p-primary ﬁnite abelian group. Then every nonidentity element of G is of order p

if and only if G ' Zp ⊕ Zp ⊕ · · · ⊕ Zp.