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- Direct_sum_of_groups abstract "In mathematics, 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 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 HThe 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 thatg = h1*h2* ... * hi * ... * hn There is a cancellation of the sum in a quotient; so that ((∑Hi) + K)/K is isomorphic to ∑HiNote the similarity with the direct product, where each g can be expressed uniquely asg = (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_sum_of_groups wikiPageID "106341".
- Direct_sum_of_groups wikiPageRevisionID "587006655".
- Direct_sum_of_groups hasPhotoCollection Direct_sum_of_groups.
- Direct_sum_of_groups subject Category:Group_theory.
- Direct_sum_of_groups comment "In mathematics, 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 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. ".
- Direct_sum_of_groups label "Direct sum of groups".
- Direct_sum_of_groups label "مجموع مباشر للزمر".
- Direct_sum_of_groups label "群的直和".
- Direct_sum_of_groups sameAs m.0qmbw.
- Direct_sum_of_groups sameAs Q4166523.
- Direct_sum_of_groups sameAs Q4166523.
- Direct_sum_of_groups wasDerivedFrom Direct_sum_of_groups?oldid=587006655.
- Direct_sum_of_groups isPrimaryTopicOf Direct_sum_of_groups.