EPFL Logo
fr|en
Login
Concept

Direct sum of groups

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
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related concepts (7)
Direct product of groups
In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted .
Direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise.
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication.
Show more

Copyright © 2025 EPFL, all rights reserved

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.