- Glossary of group theory
A

**group**("G", •) is a set "G" closed under abinary operation • satisfying the following 3axioms :* "

Associativity ": For all "a", "b" and "c" in "G", ("a" • "b") • "c" = "a" • ("b" • "c").

* "Identity element ": There exists an "e"∈"G" such that for all "a" in "G", "e" • "a" = "a" • "e" = "a".

* "Inverse element ": For each "a" in "G", there is an element "b" in "G" such that "a" • "b" = "b" • "a" = "e", where "e" is an identity element.Basic examples for groups are the

integers **Z**with addition operation, orrational numbers without zero**Q**{0} with multiplication. More generally, for any ring "R", the units in "R" form amultiplicative group . See the group article for an illustration of this definition and for further examples. Groups include, however, much more general structures than the above. Group theory is concerned with proving abstract statements about groups, regardless of the actual nature of element and the operation of the groups in question.This glossary provides short explanations of some basic notions used throughout group theory. Please refer to

group theory for a general description of the topic. See alsolist of group theory topics .**Basic definitions**A

subset "H" ⊂ "G" is a "subgroup " if the restriction of • to "H" is a group operation on "H". It is called "normal", if left and rightcoset s agree, i.e. "gH" = "Hg" for all "g" in "G". Normal subgroups play a distinguished role by virtue of the fact that the collection of cosets of a normal subgroup "N" in a group "G" naturally inherits a group structure, enabling the formation of thequotient group , usually denoted "G"/"N" (also called a "factor group"). The "Butterfly lemma " is a technical result on thelattice of subgroups of a group.Given a subset "S" of a group "G", the smallest subgroup of "G" containing "S" is called the subgroup "generated by S". It is often denoted <"S">.

Both subgroups and normal subgroups of a given group form a

complete lattice under inclusion of subsets; this property and some related results are described by thelattice theorem .Given any set "A", one can define a group as the smallest group containing the

free semigroup of $A$. This group consists of the finite strings called words that can be composed by elements from "A" and their inverses. Multiplication of strings is defined by concatenation, for instance $(abb)*(bca)=abbbca.$Every group "G" is basically a factor group of a free group generated by the set of its elements. This phenomenon is made formal with group presentations.

The "

direct product ", "direct sum", and "semidirect product " of groups glue several groups together, in different ways. The direct product of a family of groups "G"_{"i"}, for example, is thecartesian product of the sets underlying the various "G"_{"i"}, and the group operation is performed component-wise.A "

group homomorphism " is a map "f" : "G" → "H" between two groups that preserves the structure imposed by the operation, i.e. :"f"("a"•"b") = "f"("a") • "f"("b").Bijective (in-,surjective ) maps areisomorphism s of groups (mono-,epimorphism s, respectively). The kernel ker("f") is always a normal subgroup of the group. For "f" as above, the "fundamental theorem on homomorphisms " relates the structure of "G" and "H", and of the kernel and image of the homomorphism, namely:"G" / ker("f") ≅ im("f").One of the fundamental problems of group theory is the "classification of groups"

up to isomorphism.Groups together with group homomorphisms form a category.

In

universal algebra , groups are generally treated as algebraic structures of the form ("G", •, "e",^{−1}), i.e. the identity element "e" and the map that takes every element "a" of the group to its inverse "a"^{−1}are treated as integral parts of the formal definition of a group.**Finiteness conditions**The "order" |"G"| (or o("G")) of a group is the

cardinality of "G". If the order |"G"| is (in-)finite, then "G" itself is called (in-)finite. An important class is the "group of permutations" orsymmetric group s of "N" letters, denoted S_{"N"}. "Cayley's theorem " exhibits any finite group "G" as a subgroup of thesymmetric group on "G". The theory of finite groups is very rich. "Lagrange's theorem" states that the order of any subgroup "H" of a finite group "G" divides the order of "G". A partial converse is given by the "Sylow theorems ": if "p"^{"n"}is the greatest power of a prime "p" dividing the order of a finite group "G", then there exists a subgroup of order "p"^{"n"}, and the number of these subgroups is also known. Aprojective limit of finite groups is called profinite [*harvnb|Shatz|1972*] . An important profinite group, fundamental forp-adic analysis ,class field theory , andl-adic cohomology is the ring ofp-adic integers and theprofinite completion of**Z**, respectively:$mathbb\; Z\_p\; :=\; varprojlim\_n\; mathbb\; Z\; /\; p^n$ and $hat\{mathbb\; Z\}\; :=\; varprojlim\_n\; mathbb\; Z\; /\; n.$ [*These two groups play a central role for maximal*] Most of the facts from finite groups can be generalized directly to the profinite case. [abelian extension ofnumber field s, seeKronecker-Weber theorem Fact|date=June 2008*For example the Sylow theorems.Fact|date=June 2008*]Certain conditions on chains of subgroups, parallel to the notion of Noetherian and

Artinian ring s, allow to deduce further properties. For example the "Krull-Schmidt theorem " states that a group satisfying certain finiteness conditions for chains of its subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.Another, yet slightly weaker, level of finiteness is the following: a subset "A" of "G" is said to generate the group if any element "h" can be written as the product of elements of "A". A group is said to be finitely generated if it is possible to find a finite subset "A" generating the group. Finitely generated groups are in many respects as well-treatable as finite groups.

**Abelian groups**The

category of groups can be subdivided in several ways. A particularly well-understood class of groups are the so-called abelian (in honor ofNiels Abel , orcommutative ) groups, i.e. the ones satisfying:"a" • "b" = "b" • "a" for all "a", "b" in "G".Another way of saying this is that thecommutator : ["a", "b"] := "a"^{−1}"b"^{−1}"ab"equals the identity element. A "non-abelian" group is a group that is not abelian. Even more particular, cyclic groups are the groups generated by a single element. Being either isomorphic to**Z**or to**Z**_{"n"}, the integersmodulo "n", they are always abelian. Anyfinitely generated abelian group is known to be adirect sum of groups of these two types. The category of abelian groups is anabelian category . In fact, abelian groups serve as the prototype of abelian categories. A converse is given byMitchell's embedding theorem .**Normal series**Most of the notions developed in group theory are designed to tackle non-abelian groups. There are several notions designed to measure how far a group is from being abelian. The

commutator subgroup (or derived group) is the subgroup generated by commutators ["a", "b"] , whereas the center is the subgroup of elements that commute with every other group element.Given a group "G" and a normal subgroup "N" of "G", denoted "N" ⊲ "G", there is an

exact sequence ::1 → "N" → "G" → "H" → 1,where 1 denotes thetrivial group and "H" is the quotient "G"/"N". This permits the decomposition of "G" into two smaller pieces. The other way round, given two groups "N" and "H", a group "G" fitting into an exact sequence as above is called an "extension" of "H" by "N". Given "H" and "N" there are many different group extensions "G", which leads to theextension problem . There is always at least one extension, called the trivial extension, namely thedirect sum nowrap begin"G" = "N" ⊕ "H"nowrap end, but usually there are more. For example, theKlein four-group is a non-trivial extension of**Z**_{2}by**Z**_{2}. This is a first glimpse ofhomological algebra and "Ext" functors. [*harvnb|Weibel|1994*]Many properties for groups, for example being a

finite group or a "p"-group (i.e. the order of every element is a power of "p") are stable under extensions and sub- and quotient groups, i.e. if "N" and "H" have the property, then so does "G" and vice versa. This kind of information is therefore preserved while breaking it into pieces by means of exact sequences. If this process has come to an end, i.e. if a group "G" does not have any (non-trivial) normal subgroups, "G" is called simple. The name is misleading because a simple group can in fact be very complex. An example is themonster group , whose order is about 10^{54}. The finite simple groups are known and classified.Repeatedly taking normal subgroups (if they exist) leads to "

normal series "::1 = "G"_{0}⊲ "G"_{1}⊲ ... ⊲ "G"_{n}= "G",i.e. any "G"_{"i"}is a normal subgroup of the next one "G"_{"i"+1}. A group is "solvable" (or "soluble") if it has a normal series all of whose quotients are abelian. Imposing further commutativity constraints on the quotients "G"_{"i"+1}/ "G"_{"i"}, one obtainscentral series which lead to "nilpotent group s". They are an approximation of abelian groups in the sense that: [... ["g"_{1}, "g"_{2}] , "g"_{3}] ..., "g"_{"n"}] =1for all choices of group elements "g"_{"i"}.There may be distinct normal series for a group "G". If it is impossible to refine a given series by inserting further normal subgroups, it is called "

composition series ". By the "Jordan-Hölder theorem " any two composition series of a given group are equivalent. [*This can be shown using the*]Schreier refinement theorem .**Other notions**, denoted by GL("n", "F"), is the group of $n$-by-$n$ invertible matrices, where the elements of the matrices are taken from a field $F$ such as the real numbers or the complex numbers.General linear group (not to be confused with the "presentation" of a group). A "group representation" is a homomorphism from a group to a general linear group. One basically tries to "represent" a given abstract group as a concrete group of invertible matrices which is much easier to study.Group representation **References***cite book | author=Rotman, Joseph | title=An introduction to the theory of groups | location=New York | publisher=Springer-Verlag | year=1994 | id=ISBN 0-387-94285-8 A standard contemporary reference.

* | year=1994

* | year=1972**Notes**

*Wikimedia Foundation.
2010.*