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Direct sum of groups

In group theory, the direct sum of an indexed family of groups \{G_i \mid i \in I\} is a subgroup of their direct product \prod_{i \in I} G_i, consisting of all tuples (g_i)_{i \in I} where g_i is the identity element of G_i for all but finitely many i \in I, equipped with componentwise group operation.^[1]^[2] For finite index sets, the direct sum coincides with the direct product.^[1]^[2] When the groups G_i are abelian, the direct sum is often denoted \bigoplus_{i \in I} G_i and uses additive notation to reflect their structure as \mathbb{Z}-modules, distinguishing it notationally from the multiplicative direct product while preserving the same algebraic properties for finite families.^[3]^[1] This construction is central to the study of abelian groups, enabling their decomposition into simpler components; for instance, every finite abelian group is isomorphic to a direct sum of cyclic groups of prime power order, as stated in the fundamental theorem of finite abelian groups.^[2] The direct sum also admits an internal formulation: a group G is the internal direct sum of normal subgroups \{N_i \mid i \in I\} if G is generated by their union and the subgroups pairwise intersect trivially in a manner that ensures the external direct sum embeds isomorphically into G.^[1] In the category of abelian groups, the direct sum serves as the coproduct, facilitating universal properties for homomorphisms and extensions.^[1] These properties underpin applications in representation theory, module theory, and the classification of torsion groups, where infinite direct sums model structures like the quotient of rational numbers by integers as \mathbb{Q}/\mathbb{Z} \cong \bigoplus_p \mathbb{Z}[p^{-1}]/\mathbb{Z} over primes p.^[4]

Core Construction

Definition

The direct sum of groups is a construction primarily defined in the context of abelian groups, where it serves as the coproduct in the category of abelian groups; for non-abelian groups, the analogous coproduct is the free product.^[5]^[6] For a finite family of abelian groups \{G_i\}_{i=1}^n, the direct sum \bigoplus_{i=1}^n G_i is the set of n-tuples (g_1, \dots, g_n) with g_i \in G_i for each i, equipped with the componentwise group operation: (g_1, \dots, g_n) + (h_1, \dots, h_n) = (g_1 + h_1, \dots, g_n + h_n).^[7] In the finite case, this direct sum is isomorphic to the direct product \prod_{i=1}^n G_i.^[6] The notation \oplus is used for the direct sum to distinguish it from the direct product \times (or \prod), particularly in infinite cases where the two differ, though the focus here is on finite families.^[7] The direct sum satisfies the universal property of the coproduct in the category of abelian groups: given any abelian group H and group homomorphisms \phi_i: G_i \to H for i=1,\dots,n, there exists a unique group homomorphism \phi: \bigoplus_{i=1}^n G_i \to H such that the following diagrams commute for each i, where \iota_i: G_i \to \bigoplus_{j=1}^n G_j is the inclusion map sending g_i to the tuple with g_i in the i-th position and the identity elsewhere:

\begin{CD} G_i @>\iota_i>> \bigoplus_{j=1}^n G_j \\ @V{\phi_i}VV @VV{\phi}V \\ H @= H \end{CD}

That is, \phi \circ \iota_i = \phi_i for all i.^[6]^[7] To see why this construction satisfies the universal property, note that any homomorphism \phi: \bigoplus_{i=1}^n G_i \to H is determined by its compositions with the inclusions \iota_i, since elements of the direct sum are finite linear combinations (in the abelian sense) of the images of the \iota_i. Specifically, for (g_1, \dots, g_n) \in \bigoplus_{i=1}^n G_i, define \phi(g_1, \dots, g_n) = \sum_{i=1}^n \phi_i(g_i); this is well-defined because the operation is componentwise, and uniqueness follows from the fact that the inclusions generate the direct sum as a group. This yields an isomorphism \mathrm{Hom}(\bigoplus_{i=1}^n G_i, H) \cong \prod_{i=1}^n \mathrm{Hom}(G_i, H), confirming the coproduct structure.^[7]

Basic Properties

The direct sum of groups possesses several fundamental algebraic properties that follow from its construction as the subgroup of the direct product consisting of elements with finitely many nonzero components. One key property is that direct sums preserve exact sequences. Specifically, if $0 \to A_i \to B_i \to C_i \to 0 is a short exact sequence of abelian groups for each index i in a set I, then the induced sequence $0 \to \bigoplus_{i \in I} A_i \to \bigoplus_{i \in I} B_i \to \bigoplus_{i \in I} C_i \to 0 is also exact, as the componentwise maps ensure kernels and images align precisely across the sum.^[8] Subgroups formed by direct sums inherit normality from the individual components, particularly in the abelian case. If H_i \leq G_i for each i \in I, then \bigoplus_{i \in I} H_i is a normal subgroup of \bigoplus_{i \in I} G_i, since all subgroups of abelian groups are normal and the operations are componentwise. Moreover, the quotient \left( \bigoplus_{i \in I} G_i \right) / \left( \bigoplus_{i \in I} H_i \right) is isomorphic to \bigoplus_{i \in I} (G_i / H_i), reflecting the compatibility of the quotient operation with the direct sum structure.^[9] Homomorphisms between direct sums can be decomposed systematically when the index sets match. A group homomorphism \phi: \bigoplus_{i \in I} G_i \to \bigoplus_{j \in J} H_j with I = J factors into a "matrix" of individual homomorphisms \phi_{jk}: G_k \to H_j for each pair (j, k), where the image of an element in the sum is determined by applying these components and collecting nonzero terms. This decomposition arises from the universal property of the direct sum, allowing any such map to be expressed via its actions on the summands.^[8] For finite index sets, the direct sum coincides with the direct product. Explicitly, \bigoplus_{i=1}^n G_i \cong \prod_{i=1}^n G_i via the natural isomorphism that identifies tuples (g_1, \dots, g_n) in the product with their corresponding elements in the sum, as every component is nonzero in finite cases. This equivalence simplifies many computations and highlights the uniformity of the construction for bounded collections.^[8] In the context of abelian groups, additional structural properties hold for central and derived components. The center of the direct sum is the direct sum of the centers: Z\left( \bigoplus_{i \in I} G_i \right) = \bigoplus_{i \in I} Z(G_i), since commutativity with elements across components requires componentwise centrality. Similarly, the derived subgroup satisfies [\bigoplus_{i \in I} G_i, \bigoplus_{i \in I} G_i] = \bigoplus_{i \in I} [G_i, G_i], as commutators in the sum are generated componentwise due to the abelian nature of the overall structure.^[9]

Structural Components

Direct Summands

In the context of abelian groups, a subgroup H of an abelian group G is called a direct summand if there exists another subgroup K of G such that G = H \oplus K. This means that every element g \in G can be uniquely expressed as g = h + k for some h \in H and k \in K.^[10] Equivalently, H is a direct summand if there exists a projection homomorphism \pi: G \to H and an inclusion homomorphism \iota: H \to G such that \pi \circ \iota = \mathrm{id}_H. The projection \pi satisfies \pi(h) = h for all h \in H, and the inclusion \iota embeds H into G as a subgroup.^[11] A further characterization is that H is a direct summand of G if and only if the short exact sequence

$0 \to H \xrightarrow{\iota} G \to G/H \to 0

splits, meaning there exists a homomorphism \sigma: G/H \to G such that the composition of the quotient map G \to G/H with \sigma is the identity on G/H. This splitting ensures the existence of a complementary subgroup isomorphic to G/H.^[10] In categorical terms, direct summands correspond to retracts in the category of abelian groups: H is a retract of G via the pair of morphisms \iota and \pi satisfying the identity condition above. For example, the subgroup $2\mathbb{Z} of \mathbb{Z} (the even integers) is not a direct summand, as there is no subgroup P of \mathbb{Z} such that \mathbb{Z} = 2\mathbb{Z} \oplus P; any potential complement would fail to account for the generator 1 of \mathbb{Z}.^[10]

Decompositions into Direct Sums

The primary decomposition theorem states that every finitely generated abelian group G is isomorphic to a direct sum \mathbb{Z}^r \oplus \bigoplus_{i=1}^m \mathbb{Z}/p_i^{e_i}\mathbb{Z}, where r is the rank of G, the p_i are primes, and the e_i > 0 are positive integers; this decomposition into elementary divisors is unique up to isomorphism of the summands and their order.^[12] An equivalent formulation uses invariant factors, where G \cong \mathbb{Z}^r \oplus \mathbb{Z}/d_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/d_k\mathbb{Z} with d_1 \mid d_2 \mid \cdots \mid d_k and each d_i > 1; the invariant factors are also unique up to isomorphism. These two decompositions are related by grouping the primary components according to their prime powers, and the uniqueness follows from the structure of finitely generated modules over the principal ideal domain \mathbb{Z}.^[13] Two direct sum decompositions G \cong \bigoplus_{i=1}^s A_i and G \cong \bigoplus_{j=1}^t B_j of an abelian group G are equivalent if s = t, there exists a permutation \sigma of the indices such that A_i \cong B_{\sigma(i)} for each i, and the isomorphisms pair the summands accordingly. For finitely generated abelian groups, the primary decomposition theorem guarantees that all such decompositions into cyclic summands are equivalent in this sense.^[12] The Krull-Schmidt theorem asserts that if an abelian group admits a decomposition into a finite direct sum of indecomposable summands, and the group satisfies certain finiteness conditions—such as having bounded order (i.e., being a bounded torsion group) or being Noetherian (i.e., finitely generated)—then any two such decompositions are equivalent up to isomorphism and permutation of the summands.^[13] The indecomposable abelian groups under these conditions are precisely the cyclic groups \mathbb{Z}/p^k\mathbb{Z} for primes p and k \geq 1, or \mathbb{Z} in the torsion-free case.^[14] This uniqueness relies on the endomorphism rings of the indecomposables being local rings, ensuring cancellation and rigid pairing of summands.^[13] In contrast, for torsion-free abelian groups of finite rank, the Krull-Schmidt theorem fails in general, and decompositions into indecomposable summands may not be unique up to equivalence; for instance, certain such groups admit distinct non-equivalent decompositions into indecomposable summands.^[15] Such non-uniqueness arises because the endomorphism rings are not local, allowing non-trivial automorphisms that permute summands in incompatible ways.^[16] To compute these decompositions algorithmically, represent the finitely generated abelian group G as \mathbb{Z}^n / M\mathbb{Z}^n, where M is an integer matrix; the Smith normal form of M, which diagonalizes M via unimodular transformations to D = \operatorname{diag}(d_1, \dots, d_r, 0, \dots, 0) with d_i \mid d_{i+1}, yields the invariant factors d_i directly, while the elementary divisors are obtained by factoring each d_i into prime powers. This process establishes the torsion invariants and free rank n - r, confirming the decomposition without enumeration of all possibilities.^[17]

Illustrations and Extensions

Examples

A fundamental example of a finite direct sum arises in the category of abelian groups, where \mathbb{Z} \oplus \mathbb{Z} is isomorphic to \mathbb{Z}^2, the free abelian group of rank 2. Here, the elements (1,0) and (0,1) form a basis, generating the group additively while satisfying no non-trivial relations beyond commutativity. This construction illustrates how direct sums preserve the free structure, allowing independent generation by basis elements.^[6] For torsion groups, the Klein four-group V_4 = \{e, a, b, c\}, where each non-identity element has order 2 and the group is abelian, decomposes as V_4 \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}. The summands are cyclic groups generated by a and b, respectively, with c = a + b, demonstrating a direct sum into indecomposable cyclic components of prime order. This decomposition highlights the role of direct sums in classifying small non-cyclic groups.^[6] Another illustration from finitely generated abelian groups uses the Chinese Remainder Theorem: since 6 = 2 \cdot 3 with \gcd(2,3)=1, the ring isomorphism \mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} induces a group isomorphism \mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z}. The summands correspond to the primary components, with generators of orders 2 and 3, respectively, and their direct sum captures the full structure without overlap.^[18] In contrast, the rational numbers \mathbb{Q} viewed as a \mathbb{Z}-module is indecomposable because it is torsion-free of rank 1; the rank is additive under direct sums, so it cannot decompose into nontrivial direct summands.^[19] Direct sums also appear in applications to topology, such as the classification of homology groups. For the n-dimensional torus T^n, the first singular homology group is H_1(T^n) \cong \mathbb{Z}^n = \bigoplus_{i=1}^n \mathbb{Z}, reflecting the n independent 1-cycles from the fundamental group's abelianization. This decomposition aids in understanding abelian extensions and cohomological invariants of manifolds.^[20] Uniqueness of direct sum decompositions fails in infinite cases; for instance, the direct sum \bigoplus_{i \in I} \mathbb{Z}/2\mathbb{Z} over a countably infinite index set I admits non-unique decompositions into direct summands, as it can be rearranged or complemented in ways not possible for finite sums, violating the invariance seen in finite abelian group classifications.^[21]

Infinite Direct Sums

When the index set I is infinite, the direct sum \bigoplus_{i \in I} G_i of a family of abelian groups \{G_i \mid i \in I\} consists of elements with finite support in the direct product, as defined earlier.^[22]^[23] This construction satisfies an adapted universal property in the category of abelian groups: for any abelian group H, the group homomorphisms \operatorname{Hom}(\bigoplus_{i \in I} G_i, H) are in natural bijection with the direct product \prod_{i \in I} \operatorname{Hom}(G_i, H), where each such family of homomorphisms extends uniquely to the direct sum because elements have finite support and thus any homomorphism factors through a finite partial direct sum.^[24] A key distinction from the infinite direct product arises in the structure of elements: while the direct product \prod_{i \in I} G_i includes tuples with potentially infinite support (arbitrarily many nonzero components), the direct sum restricts to finite support, leading to significant differences in cardinality and topology. For instance, the infinite direct sum \bigoplus_{n=1}^\infty \mathbb{Z}/2\mathbb{Z} consists of all sequences in (\mathbb{Z}/2\mathbb{Z})^\mathbb{N} with only finitely many 1's, making it countable, whereas the direct product \prod_{n=1}^\infty \mathbb{Z}/2\mathbb{Z} is uncountable with cardinality $2^{\aleph_0}.^[22]^[23] Infinite direct sums inherit several important properties from their summands. If each G_i is a free abelian group, then \bigoplus_{i \in I} G_i is also free abelian, with basis given by the disjoint union of bases for the G_i. Moreover, when I is countably infinite, such direct sums are slender abelian groups, meaning they do not admit certain homomorphisms from \mathbb{Z}^\mathbb{N} other than those factoring through finite partial products; specifically, any group homomorphism \phi: \mathbb{Z}^\mathbb{N} \to \bigoplus_{n=1}^\infty \mathbb{Z} has \phi(e_n) = 0 for all but finitely many standard basis elements e_n.^[23] This construction generalizes naturally to the category of modules over a ring R, where the direct sum \bigoplus_{i \in I} M_i of R-modules is the R-submodule of the direct product consisting of elements with finite support, serving as the coproduct in that category and linking directly to the broader theory of modules without altering the finite-support condition. In number theory, a related notion appears in the form of restricted direct products, such as the adele ring \mathbb{A}_F of a number field F, which is the restricted product over all places v of the completions F_v, comprising elements (a_v) where a_v \in \mathcal{O}_v (the ring of integers at v) for all but finitely many v; this mirrors the finite-support restriction and endows the structure with a locally compact topology.^[25]^[26]

References

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[PDF] Section I.8. Direct Products and Direct Sums
Nov 22, 2023 · So the external weak direct product (or external direct sum) is not in general a coproduct in the category of all groups (as is to be shown in ...
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It is easy to check that this is a group. If G and H are abelian we often call their direct product the direct sum and denote it by G H.
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Gk is a product in the category of groups. In the case that each group ... a coproduct in the category of abelian groups. In the case that the groups ...
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Below is a merged summary of the direct sum of groups or modules as presented in Serge Lang's *Algebra* (Revised Third Edition), based on the provided segments. To retain all information in a dense and organized manner, I will use a table in CSV format for the detailed properties across different sections, followed by a narrative summary that consolidates additional details, quotes, and URLs. This approach ensures comprehensive coverage while maintaining clarity.
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[PDF] Abstract Algebra
... direct or Cartesian product of A and B. H is a subgroup of G the cyclic ... abstract algebra begins with the disarmingly simple abstract definition of a ...
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When a Short Exact Sequence Splits. Theorem 2.1. Let 0 −→ N f. −−→ M g ... Using Theorem 2.1, we can describe when a submodule N ⊂ M is a direct summand.
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Direct Summand -- from Wolfram MathWorld
Given the direct sum of additive Abelian groups A direct sum B, A and B are called direct summands. The map i_1:A-->A direct sum B defined by i(a)=a direct ...
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Kronecker Decomposition Theorem -- from Wolfram MathWorld
Kronecker Decomposition Theorem. Every finite Abelian group can be written as a group direct product of cyclic groups of prime power group orders. In fact, the ...Missing: primary | Show results with:primary
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The Krull-Schmidt theorem - ScienceDirect
The “classical” Krull-Schmidt theorem asserts that any two direct sum decompositions of a module of finite length into indecomposable summands are isomorphic.
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Jan 5, 2024 · The Krull-Schmidt Theorem deals with writing certain types of groups as products of indecomposable groups. The “certain type” involves the ...
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Jan 10, 2017 · Most published examples of groups with non–unique decompositions are almost completely decomposable groups. It is also noteworthy that for an ...
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Jul 7, 2021 · This article deals with the formalisation of some group-theoretic results including the fundamental theorem of finitely generated abelian ...
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[PDF] The Fundamental Theorem for Finite Abelian Groups
In the abelian case, the direct sum and the direct product of finitely many groups coincide. The sum of non-abelian groups is much more difficult to deal with ...<|control11|><|separator|>
[18]
[PDF] Modules and Vector Spaces - LSU Math
However, if S is complemented, then a complement T ∼= Q; so Q is a submodule of a free Z-module M, and hence Q would be free, but Q is not a free Z-module.
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Uniqueness of infinite direct sum decomposition - MathOverflow
Dec 13, 2020 · If M admits a finite decomposition M=⨁ni=1Si into simple R-modules Si, then this decomposition is unique up to isomorphism and permutation of ...How complicated is infinite-dimensional "undergraduate linear ...Extensions of an infinite product of copies of Z by Z - MathOverflowMore results from mathoverflow.net
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Infinite direct sum of abelian groups - Math Stack Exchange
Jun 4, 2016 · It is simply the subgroup of all elements (ai) in the product ∏i∈ZAi such that all ai are 0 but a finite number.Missing: "Fuchs" | Show results with:"Fuchs"<|control11|><|separator|>
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Author, László Fuchs ; Publisher, Academic Press, 1970 ; Original from, University of Illinois at Urbana-Champaign ; Digitized, Jul 31, 2019.
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[PDF] LECTURE 5 (5.0) Review.– Recall that last time we ... - OSU Math
inclusions). II. Ab : In the category of abelian groups, the direct sum and product are given as follows: Ab. M i∈I. Gi = {(xi ∈ Gi)i∈I : gi = 0 for all ...
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Nov 20, 2023 · An important special case is when the Xλ are topological Abelian groups or rings (cf. Topological group, Topological ring). Assume that the ...
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Restricted Direct Products in Koch's Number Theory
May 8, 2011 · Such "restricted direct products" are colimits of the subgroups GS where all but the factors in S are in the subgroups Hi.