Direct sum of groups
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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[edit]
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 =
Properties[edit]
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 =
- 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 ((
Σ Hi) + K)/K is isomorphic toΣ Hi.
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,
Direct summand[edit]
Given a group , we say that a subgroup is a direct summand of if there exists another subgroup of such that .
In abelian groups, if is a divisible subgroup of , then is a direct summand of .
Examples[edit]
- 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 .
Equivalence of decompositions into direct sums[edit]
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 =
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[edit]
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
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 =