A group G is called the direct sum[1][2] of two subgroups H1 and H2 if
- each H1 and H2 are normal subgroups of G,
- the subgroups H1 and H2 have trivial intersection (i.e., having only the identity element of G in common),
- G = ⟨H1, H2⟩; in other words, G is generated by the subgroups H1 and H2.
More generally, G is called the direct sum of a finite set of subgroups {Hi} if
- each Hi is a normal subgroup of G,
- each Hi has trivial intersection with the subgroup ⟨{Hj : j ≠ i}⟩,
- G = ⟨{Hi}⟩; in other words, G is generated by the subgroups {Hi}.
If G is the direct sum of subgroups H and K then we write G = H + K, and if G is the direct sum of a set of subgroups {Hi} then we often write G = ΣHi. Loosely speaking, a direct sum is isomorphic to a weak direct product of subgroups.
If G = H + K, then it can be proven that:
- for all h in H, k in K, we have that h ∗ k = k ∗ h
- for all g in G, there exists unique h in H, k in K such that g = h ∗ k
- There is a cancellation of the sum in a quotient; so that (H + K)/K is isomorphic to H
The above assertions can be generalized to the case of G = ΣHi, where {Hi} is a finite set of subgroups:
- if i ≠ j, then for all hi in Hi, hj in Hj, we have that hi ∗ hj = hj ∗ hi
- for each g in G, there exists a unique set of elements hi in Hi such that
- g = h1 ∗ h2 ∗ ... ∗ hi ∗ ... ∗ hn
- There is a cancellation of the sum in a quotient; so that ((ΣHi) + K)/K is isomorphic to ΣHi.
Note the similarity with the direct product, where each g can be expressed uniquely as
- g = (h1,h2, ..., hi, ..., hn).
Since hi ∗ hj = hj ∗ hi for all i ≠ j, it follows that multiplication of elements in a direct sum is isomorphic to multiplication of the corresponding elements in the direct product; thus for finite sets of subgroups, ΣHi is isomorphic to the direct product ×{Hi}.
- If we take it is clear that is the direct product of the subgroups .
- If is a divisible subgroup of an abelian group then there exists another subgroup of such that .
- If also has a vector space structure then can be written as a direct sum of and another subspace that will be isomorphic to the quotient .
In the decomposition of a finite group into a direct sum of indecomposable subgroups the embedding of the subgroups is not unique. For example, in the Klein group we have that
- and
However, the Remak-Krull-Schmidt theorem states that given a finite group G = ΣAi = ΣBj, where each Ai and each Bj is non-trivial and indecomposable, the two sums have equal terms up to reordering and isomorphism.
The Remak-Krull-Schmidt theorem fails for infinite groups; so in the case of infinite G = H + K = L + M, even when all subgroups are non-trivial and indecomposable, we cannot conclude that H is isomorphic to either L or M.
To describe the above properties in the case where G is the direct sum of an infinite (perhaps uncountable) set of subgroups, more care is needed.
If g is an element of the cartesian product Π{Hi} of a set of groups, let gi be the ith element of g in the product. The external direct sum of a set of groups {Hi} (written as ΣE{Hi}) is the subset of Π{Hi}, where, for each element g of ΣE{Hi}, gi is the identity for all but a finite number of gi (equivalently, only a finite number of gi are not the identity). The group operation in the external direct sum is pointwise multiplication, as in the usual direct product.
This subset does indeed form a group, and for a finite set of groups {Hi} the external direct sum is equal to the direct product.
If G = ΣHi, then G is isomorphic to ΣE{Hi}. Thus, in a sense, the direct sum is an "internal" external direct sum. For each element g in G, there is a unique finite set S and a unique set {hi ∈ Hi : i ∈ S} such that g = Π {hi : i in S}.
Homology. Saunders MacLane. Springer, Berlin; Academic Press, New York, 1963.
László Fuchs. Infinite Abelian Groups