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Concept
Direct sum of groups
Summary
In mathematics, a group G is called the direct sum 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.
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 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).
Official source
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