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Direct sum of groups

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Direct sum of groups

In mathematics, a group G is called the direct sum of two subgroups H1 and H2 if

Contents

  • each H1 and H2 are normal subgroups of G
  • the subgroups H1 and H2 have trivial intersection (i.e., having only the identity element e in common), and
  • 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 not equal to i}>, 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 i ≠ j, 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
    • g = h1*h2* ... * hi * ... * hn
  • 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

    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, ∑Hi is isomorphic to the direct product ×{Hi}.

    Direct summand

    Given a group G , we say that a subgroup H is a direct summand of G (or that splits form G ) if and only if there exist another subgroup K ≤ G such that G is the direct sum of the subgroups H and K

    In abelian groups, if H is a divisible subgroup of G then H is a direct summand of G .

    Examples

  • If we take
    • G = ∏ i ∈ I H i it is clear that G is the direct product of the subgroups H i 0 × ∏ i ≠ i 0 H i .
  • If H is a divisible subgroup of an abelian group G . Then there exist another subgroup K ≤ G such that G = K + H
  • If G is also a vector space then G can be written as a direct sum of R and another subespace K that will be isomorphic to the quotient 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, V4 = C2 × C2, we have that

    V4 = <(0,1)> + <(1,0)> andV4 = <(1,1)> + <(1,0)>.

    However, it is the content of the Remak-Krull-Schmidt theorem that given a finite group G = ∑Ai = ∑Bj, where each Ai and each Bj 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 then assume 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 ∏{Hi} of a set of groups, let gi be the ith element of g in the product. The external direct sum of a set of groups {Hi} (written as ∑E{Hi}) is the subset of ∏{Hi}, where, for each element g of ∑E{Hi}, gi is the identity e H i for all but a finite number of gi (equivalently, only a finite number of gi 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 Hi, the external direct sum is identical to the direct product.

    If G = ∑Hi, then G is isomorphic to ∑E{Hi}. 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 unique {hi in Hi : i in S} such that g = ∏ {hi : i in S}.

    References

    Direct sum of groups Wikipedia
    (Text) CC BY-SA


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