The structure theorem for finite abelian groups is most conveniently obtained in the context of studying finitely-generated modules over the ring of integers, or, more generally, over principal ideal domains.
On the other hand, the segment of this course on the topic of finite groups is hardly a complete introduction to that subject without at least mention of the structure theorem for finite abelian groups, and it is desirable to have a treatment that fits the context of this course.
The method used here, explained to me by Professor Alexandre Tchernev, is based on analysis of the case of a finite abelian group of prime power order in the spirit of “Nakayama's Lemma” from the subject of commutative algebra.
The structure theorem for finite abelian groups is the following:
Z/m_{1}Z \times Z/m_{2}Z \times … \times Z/m_{r}Z |
1 <= m_{1} | m_{2} | … | m_{r} . |
In view of the Chinese Remainder Theorem, i.e.,
gcd(m, n) = 1 ⇒ Z/mnZ ~= Z/mZ \times Z/nZ , |
this theorem is equivalent to the following:
The proof will proceed in stages.
Proof. If G is a given finite abelian group, then each of its Sylow p-subgroups must be normal in G since every subgroup of an abelian group is normal. Since for a given prime p all Sylow p-subgroups are conjugate to each other, it follows that for each prime p dividing the order of G there is one and only one Sylow p-subgroup H_{p}. Since these subgroups are all normal the map
PROD_{p}[ H_{p} ] \overset{varphi}{ ----> } G |
given by varphi(h_{1}, h_{2}, …, h_{r}) = h_{1} h_{2} … h_{r}, where h_{i} \in H_{p_{i}} for the distinct primes p_{1}, p_{2}, …, p_{r} dividing |G| , must be a group homomorphism. Since the orders of the groups H_{p_{i}} are pairwise coprime, the homomorphism varphi must be injective, and since the domain and target of varphi both have the same number of elements, varphi must, since injective, be bijective. Thus G is isomorphic via varphi to the direct product of its Sylow subgroups.
Example. The exponent of the symmetric group S_{4} (which has order 24) is 12. The largest order of any element in S_{4} is 4.
Proof. In view of the preceding proposition the question reduces to the case of a finite abelian group having prime power order. In that case the statement is obvious.
To prove theorem 2 it suffices to prove that a finite group with order a power of p, where p is prime, is isomorphic to the direct product of cyclic groups. If A is a finite abelian group of order |A| = p^{m}, with group law written additively, then the set pA = {p x | x \in A} is a subgroup of A, and \={A} will denote the quotient
\={A} = A/pA . |
Proof. If A is non-trivial, then its exponent must be p^{k} for some k >= 1, and, clearly, the exponent of pA is then p^{k-1}. Since A and pA have different exponents, they cannot be equal.
Example. If e_{1}, …, e_{r} are integers with e_{j} >= 1 for 1 <= j <= r, then
A ~= Z/p^{e_{1}}Z \times … Z/p^{e_{r}}Z ⇒ \={A} ~= |
| ^{r} . |
In this example note that if x_{j} is a generator of the j-th factor Z/p^{e_{j}}Z and \={x}_{j} = pi(x_{j}), where pi is the quotient homomorphism from A to \={A}, then \={x}_{1}, …, \={x}_{r} generate \={A}. Moreover, \={A} is a vector space over the field Z/pZ, and \={x}_{1}, …, \={x}_{r} is a basis of \={A}. Unfortunately, showing that a finite abelian group A with order p^{m} is isomorphic to a direct product of cyclic groups is a bit more complicated than finding a basis of \={A} as a vector space over Z/pZ.
Proof. Let H be the subgroup of G generated by x_{1}, …, x_{r}, and let pi : G --> \={G} be the quotient homorphism. Let varphi be the homomorphism obtained by following the quotient homomorphism G --> G/H with the quotient homomorphism G/H --> \overline{(G/H)}. Clearly pG is contained in the kernel of varphi, hence, by the universal mapping property for quotients one obtains a homomorphism \={varphi} : \={G} --> \overline{(G/H)}. The image of \={varphi} is the same as the image of varphi, and, therefore, is \overline{(G/H)} since varphi is the composition of two surjective homomorphisms. The kernel of \={varphi} is the image under pi of the kernel of varphi. The kernel of varphi is the subgroup H + pG of G generated by the set H cup pG. Since pi(pG) = (0), the kernel of \={varphi} is simply pi(H). But pi(H) = \={G} by hypothesis, and, therefore
\overline{(G/H)} = Image(\={varphi}) ~= (0) , |
i.e., p(G/H) = G/H. Hence, G/H ~= (0), i.e., G is generated by x_{1}, …, x_{r}.
Proof. The first of these corollaries is obvious from what was previously shown. The second follows from the fact that any linearly independent set in a finite-dimensional vector space over a field is part of some basis of that vector space.
Example. For the group
Z/2Z \times Z/4Z |
the set
{ | (1, 1), (0, 1) | } |
Z/p^{e_{1}}Z \times … \times Z/p^{e_{r}}Z |
Proof. When x_{1}, …, x_{r} is a minimal set of generators of G with x_{j} having order p^{e_{j}}, it will always be assumed that the sequence is arranged in such a way that the orders increase, i.e., e_{1} <= … <= e_{r}. Among all so-arranged minimal sets of generators of G choose one for which the sequence of orders is lexicographically smallest. (Note that this is not the case for the particular minimal generating set in the example given above following corollary 9.) Define the group homomorphism varphi : Z^{r} --> G by the formula
varphi(n_{1}, …, n_{r}) = n_{1} x_{1} + … + n_{r} x_{r} . |
Clearly, varphi is surjective since x_{1}, …, x_{r} generate G. The proof of the proposition will have been obtained if it is shown that (n_{1}, …, n_{r}) \in Ker(varphi) if and only if n_{j} EQUIV 0 (mod p^{e_{j}}) for 1 <= j <= r. Clearly, the simultaneous congruences are sufficient for membership in the kernel. Suppose now that (n_{1}, …, n_{r}) is in the kernel, but at least one of the coordinates n_{j} does not satisfy the desired congruence. Suppose s is the smallest such value of the index j. Then
n_{s} x_{s} + … + n_{r} x_{r} = 0 |
and, therefore,
n_{s} \={x}_{s} + … + n_{r} \={x}_{r} = 0 . |
Since \={x}_{s}, …, \={x}_{r} are linearly independent over the field Z/pZ, it follows that n_{j} EQUIV 0 (mod p) for the indices j with s <= j <= r. Let p^{k} be the highest power of p dividing all of the integers n_{s}, …, n_{r}, and let n_{j} = p^{k} m_{j} for s <= j <= r. Let y = m_{s} x_{s} + … + m_{r} x_{r}. By the choice of the exponent k one has m_{j} EQUIV0 (mod p) for some j, s <= j <= r. Since \={x}_{s}, …, \={x}_{r} are linearly independent over Z/pZ, one finds that \={y} = m_{s} \={x}_{s} + … + m_{r} \={x}_{r} ≠ 0. Hence, the set \={x}_{1}, …, \={x}_{s-1}, \={y} is linearly independent, and, therefore, by corollary 9, may be completed to a basis \={x}_{1}, …, \={x}_{s-1}, \={y}_{s}, …, \={y}_{r} of \={G} where y_{s} = y. Since n_{s} = p^{k} m_{s} EQUIV0 (mod p^{e_{s}}), one sees that 1 <= k < e_{s}, which is to say that the set x_{1}, …, x_{s-1}, y_{s}, …, y_{r} -- which is a minimal generating set by proposition 7 -- is lexicographically smaller than x_{1}, …, x_{r} contrary to the choice of x_{1}, …, x_{r}.