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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Krull–Schmidt theorem states that a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
subjected to certain finiteness conditions on
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
s of
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
s, can be uniquely written as a finite
direct product In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...
of indecomposable subgroups.


Definitions

We say that a group ''G'' satisfies the
ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings. These conditions p ...
(ACC) on subgroups if every
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of subgroups of ''G'': :1 = G_0 \le G_1 \le G_2 \le \cdots\, is eventually constant, i.e., there exists ''N'' such that ''G''''N'' = ''G''''N''+1 = ''G''''N''+2 = ... . We say that ''G'' satisfies the ACC on normal subgroups if every such sequence of normal subgroups of ''G'' eventually becomes constant. Likewise, one can define the descending chain condition on (normal) subgroups, by looking at all decreasing sequences of (normal) subgroups: :G = G_0 \ge G_1 \ge G_2 \ge \cdots.\, Clearly, all finite groups satisfy both ACC and DCC on subgroups. The
infinite cyclic group In abstract algebra, a cyclic group or monogenous group is a group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of -adic numbers), that is generated by a single element. That is, it is a set of invertib ...
\mathbf satisfies ACC but not DCC, since (2) > (2)2 > (2)3 > ... is an infinite decreasing sequence of subgroups. On the other hand, the p^\infty-torsion part of \mathbf/\mathbf (the quasicyclic ''p''-group) satisfies DCC but not ACC. We say a group ''G'' is indecomposable if it cannot be written as a direct product of non-trivial subgroups ''G'' = ''H'' × ''K''.


Statement

If G is a group that satisfies both ACC and DCC on normal subgroups, then there is exactly one way of writing G as a direct product G_1 \times G_2 \times\cdots \times G_k\, of finitely many indecomposable subgroups of G. Here, uniqueness means direct decompositions into indecomposable subgroups have the exchange property. That is: suppose G = H_1 \times H_2 \times \cdots \times H_l\, is another expression of G as a product of indecomposable subgroups. Then k=l and there is a reindexing of the H_i's satisfying * G_i and H_i are isomorphic for each i; * G = G_1 \times \cdots \times G_r \times H_ \times\cdots\times H_l\, for each r.


Proof

Proving existence is relatively straightforward: let be the set of all normal subgroups that can not be written as a product of indecomposable subgroups. Moreover, any indecomposable subgroup is (trivially) the one-term direct product of itself, hence decomposable. If Krull-Schmidt fails, then contains ; so we may iteratively construct a descending series of direct factors; this contradicts the DCC. One can then invert the construction to show that ''all'' direct factors of appear in this way. The proof of uniqueness, on the other hand, is quite long and requires a sequence of technical lemmas. For a complete exposition, see.


Remark

The theorem does not assert the existence of a non-trivial decomposition, but merely that any such two decompositions (if they exist) are the same.


Remak decomposition

A Remak decomposition, introduced by Robert Remak, is a decomposition of an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
or similar object into a finite
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. As an example, the direct sum of two abelian groups A and B is anothe ...
of indecomposable objects. The Krull–Schmidt theorem gives conditions for a Remak decomposition to exist and for its factors to be unique.


Krull–Schmidt theorem for modules

If E \neq 0 is a module that satisfies the ACC and DCC on submodules (that is, it is both
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
and Artinian or – equivalently – of finite
length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
), then E is a
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. As an example, the direct sum of two abelian groups A and B is anothe ...
of
indecomposable module In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. Indecomposable is a weaker notion than simple module (which is also sometimes called irreducible module): simple ...
s. Up to a permutation, the indecomposable components in such a direct sum are uniquely determined up to isomorphism. In general, the theorem fails if one only assumes that the module is Noetherian or Artinian.


History

The present-day Krull–Schmidt theorem was first proved by
Joseph Wedderburn Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a fi ...
(''Ann. of Math'' (1909)), for finite groups, though he mentions some credit is due to an earlier study of G.A. Miller where direct products of abelian groups were considered. Wedderburn's theorem is stated as an exchange property between direct decompositions of maximum length. However, Wedderburn's proof makes no use of automorphisms. The thesis of Robert Remak (1911) derived the same uniqueness result as Wedderburn but also proved (in modern terminology) that the group of central automorphisms acts transitively on the set of direct decompositions of maximum length of a finite group. From that stronger theorem Remak also proved various corollaries including that groups with a trivial center and perfect groups have a unique Remak decomposition. Otto Schmidt (''Sur les produits directs, S. M. F. Bull. 41'' (1913), 161–164), simplified the main theorems of Remak to the 3 page predecessor to today's textbook proofs. His method improves Remak's use of idempotents to create the appropriate central automorphisms. Both Remak and Schmidt published subsequent proofs and corollaries to their theorems.
Wolfgang Krull Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject. Krull was born and went to school in Baden-Baden. H ...
(''Über verallgemeinerte endliche Abelsche Gruppen, M. Z. 23'' (1925) 161–196), returned to G.A. Miller's original problem of direct products of abelian groups by extending to abelian operator groups with ascending and descending chain conditions. This is most often stated in the language of modules. His proof observes that the idempotents used in the proofs of Remak and Schmidt can be restricted to module homomorphisms; the remaining details of the proof are largely unchanged. O. Ore unified the proofs from various categories include finite groups, abelian operator groups, rings and algebras by proving the exchange theorem of Wedderburn holds for modular lattices with descending and ascending chain conditions. This proof makes no use of idempotents and does not reprove the transitivity of Remak's theorems. Kurosh's ''The Theory of Groups'' and Zassenhaus' ''The Theory of Groups'' include the proofs of Schmidt and Ore under the name of Remak–Schmidt but acknowledge Wedderburn and Ore. Later texts use the title Krull–Schmidt ( Hungerford's Algebra) and Krull–Schmidt– Azumaya (Curtis–Reiner). The name Krull–Schmidt is now popularly substituted for any theorem concerning uniqueness of direct products of maximum size. Some authors choose to call direct decompositions of maximum-size Remak decompositions to honor his contributions.


See also

* Krull–Schmidt category


References


Further reading

* A. Facchini: ''Module theory. Endomorphism rings and direct sum decompositions in some classes of modules.'' Progress in Mathematics, 167. Birkhäuser Verlag, Basel, 1998. * C.M. Ringel: ''Krull–Remak–Schmidt fails for Artinian modules over local rings.'' Algebr. Represent. Theory 4 (2001), no. 1, 77–86.


External links


Page at PlanetMath
{{DEFAULTSORT:Krull-Schmidt theorem Module theory Theorems in group theory