Direct sum of groups

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In mathematics, a group G is called the direct sum of a set of subgroups {Hi} if

  • each Hi is a normal subgroup of G
  • each distinct pair of subgroups has trivial intersection, and
  • 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; if G is the direct sum of a set of subgroups {Hi}, we often write G = ∑Hi. Loosely speaking, a direct sum is isomorphic to a weak direct product of subgroups.

In abstract algebra, this method of construction can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information.

This notation is commutative; so that in the case of the direct sum of two subgroups, G = H + K = K + H. It is also associative in the sense that if G = H + K, and K = L + M, then G = H + (L + M) = H + L + M.

A group which can be expressed as a direct sum of non-trivial subgroups is called decomposable; otherwise it is called indecomposable.

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 = ∑Hi, where {Hi} is a finite set of subgroups.

  • if ij, then for all hi in Hi, hj in Hj, we have that hi * hj = hj * hi
  • for each g in G, there unique set of {hi in Hi} such that
  • There is a cancellation of the sum in a quotient; so that ((∑Hi) + K)/K is isomorphic to ∑Hi

Note the similarity with the direct product, where each g can be expressed uniquely as

Since hi * hj = hj * hi for all ij, 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, ∑Hi is isomorphic to the direct product ×{Hi}.

Contents

Equivalence of direct sums

The direct sum is not unique for a group; for example, in the Klein group, V4 = C2 × C2, we have that

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