Direct sum of groups

In mathematics, a group G is called the direct sum ^{[1] [2]} of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information. A group which can be expressed as a direct sum of non-trivial subgroups is called decomposable, and if a group cannot be expressed as such a direct sum then it is called indecomposable.

Definition

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 ${\displaystyle e}$ 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 G = H + K, 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 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 = ΣH_i, where {H_i} 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 h_i ∗ h_j = h_j ∗ h_i
• 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 h_i ∗ h_j = h_j ∗ h_i 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 ${\displaystyle G}$, we say that a subgroup ${\displaystyle H}$ is a direct summand of ${\displaystyle G}$ if there exists another subgroup ${\displaystyle K}$ of ${\displaystyle G}$ such that ${\displaystyle G=H+K}$.

In abelian groups, if ${\displaystyle H}$ is a divisible subgroup of ${\displaystyle G}$, then ${\displaystyle H}$ is a direct summand of ${\displaystyle G}$.

Examples

• If we take ${\textstyle G=\prod _{i\in I}H_{i}}$ it is clear that ${\displaystyle G}$ is the direct product of the subgroups ${\textstyle H_{i_{0}}\times \prod _{i\not =i_{0}}H_{i}}$.

• If ${\displaystyle H}$ is a divisible subgroup of an abelian group ${\displaystyle G}$ then there exists another subgroup ${\displaystyle K}$ of ${\displaystyle G}$ such that ${\displaystyle G=K+H}$.
• If ${\displaystyle G}$ also has a vector space structure then ${\displaystyle G}$ can be written as a direct sum of ${\displaystyle \mathbb {R} }$ and another subspace ${\displaystyle K}$ that will be isomorphic to the quotient ${\displaystyle G/K}$.

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 ${\displaystyle V_{4}\cong C_{2}\times C_{2}}$ we have that

${\displaystyle V_{4}=\langle (0,1)\rangle +\langle (1,0)\rangle ,}$ and

${\displaystyle V_{4}=\langle (1,1)\rangle +\langle (1,0)\rangle .}$

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 Σ_E{H_i}) is the subset of Π{H_i}, where, for each element g of Σ_E{H_i}, g_i is the identity ${\displaystyle e_{H_{i}}}$ 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 Σ_E{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}.

See also

• Direct sum
• Coproduct
• Free product
• Direct sum of topological groups

References

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