Direct sum of groups

Definition

A group G is called the direct sum 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 e 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 {H**i} if

• each H**i is a normal subgroup of G,
• each H**i has trivial intersection with the subgroup ⟨{H**j : j ≠ i}⟩,
• G = ⟨{H**i}⟩; in other words, G is generated by the subgroups {H**i}.

If G is the direct sum of subgroups H and K then we write , and if G is the direct sum of a set of subgroups {H**i} then we often write G = ΣH**i. Loosely speaking, a direct sum is isomorphic to a weak direct product of subgroups.

Properties

If , then it can be proven that:

• for all h in H, k in K, we have that
• for all g in G, there exists unique h in H, k in K such that
• 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 , where {Hi} is a finite set of subgroups:

• if i ≠ j, then for all h**i in H**i, h**j in H**j, we have that
• for each g in G, there exists a unique set of elements h**i in H**i such that :g = h1 ∗ h2 ∗ ... ∗ h**i ∗ ... ∗ h**n
• There is a cancellation of the sum in a quotient; so that ((ΣH**i) + K)/K is isomorphic to ΣH**i.

Note the similarity with the direct product, where each g can be expressed uniquely as :g = (h1,h2, ..., h**i, ..., h**n).

Since 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, ΣH**i is isomorphic to the direct product ×{H**i}.

Direct summand

Given a group G, we say that a subgroup H is a direct summand of G if there exists another subgroup K of G such that G = H+K.

In abelian groups, if H is a divisible subgroup of G, then H is a direct summand of G.

Examples

• If we take G= \prod_{i\in I} H_i it is clear that G is the direct product of the subgroups H_{i_0} \times \prod_{i\not=i_0}H_i.

• If H is a divisible subgroup of an abelian group G then there exists another subgroup K of G such that G=K+H.

• If G also has a vector space structure then G can be written as a direct sum of \mathbb R and another subspace K that will be isomorphic to the quotient G/K.

Equivalence of decompositions into direct sums

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 V_4 \cong C_2 \times C_2 we have that : V_4 = \langle(0,1)\rangle + \langle(1,0)\rangle, and : V_4 = \langle(1,1)\rangle + \langle(1,0)\rangle.

However, the Remak-Krull-Schmidt theorem states that given a finite group G = ΣA**i = ΣB**j, where each A**i and each B**j 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.

Generalization to sums over infinite sets

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{H**i} of a set of groups, let g**i be the ith element of g in the product. The external direct sum of a set of groups {H**i} (written as ΣE{H**i}) is the subset of Π{H**i}, where, for each element g of ΣE{H**i}, g**i is the identity e_{H_i} for all but a finite number of g**i (equivalently, only a finite number of g**i 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 {H**i} the external direct sum is equal to the direct product.

If G = ΣH**i, then G is isomorphic to ΣE{H**i}. 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 {h**i ∈ H**i : i ∈ S} such that g = Π {h**i : i in S}.

References

References

1. Homology. Saunders MacLane. Springer, Berlin; Academic Press, New York, 1963.
2. László Fuchs. Infinite Abelian Groups