Group theory

Cayley's chart of the group free of order two.

In abstract algebra, group theory studies the algebraic structure known as a group, which is a non-empty set endowed with an internal operation. Its objectives are, among others, the classification of groups, the study of their properties and their applications both inside and outside of mathematics.

The order of a group is its cardinality; On the basis of it, groups can be classified into groups of finite order or of infinite order. The classification of simple groups of finite order is one of the greatest mathematical achievements of the XX century.

History

The historical roots of group theory are the theory of algebraic equations, number theory, and geometry. Euler, Gauss, Lagrange, Abel and Galois were the creators who laid the foundations of this branch of abstract algebra. Galois is recognized as the first mathematician who related this theory to the theory of fields, from which the Galois theory arose. In addition, he used the group name or & # 34; he invented the term [...] & # 34; according to E.T. Bell. Other major contributing mathematicians include Cayley, Emil Artin, Emmy Noether, Peter Ludwig Mejdell Sylow, A.G. Kurosch, Iwasawa among many others. It was Walter Dick who, in 1882, gave the modern definition of a group and was "the first to define the free group generated by a finite number of generators", according to Nicolás Bourbaki. At the end of the XIX century, Frobenius defined the notion of an abstract group with a system of axioms.

Group definition

A group (G, ){displaystyle (G,circ)} It's a set. G{displaystyle G} defining an internal binary operation {displaystyle circ }which satisfies the following axioms:

1. Association: Русский Русский a,b,c한 한 G:a (b c)=(a b) c{displaystyle forall a,b,cin G:acirc (bcirc c)=(acirc b)circ c}
2. Neutral element: consuming consuming e한 한 G:e a=a e=a{displaystyle exists ein G:ecirc a=acirc e=a}
3. Symmetrical element: Русский Русский a한 한 Gconsuming consuming a− − 1한 한 G:a a− − 1=a− − 1 a=e{displaystyle forall ain Gquad exists a^{-1}in G:acirc a^{-1}=a^{-1}circ a=e}

The binary operation of the group, also called Law on internal composition, specify how to compose two elements to obtain a third. It may also be considered investment as the operation would unite to each element g{displaystyle g} it makes it correspond to its reverse element g− − 1{displaystyle g^{-1}.

A group is said to be abelian or commutative when it also verifies the commutative property:

a b=b aРусский Русский a,b한 한 G{displaystyle acirc b=bcirc aquad forall a,bin G}

Notation

You talk about it. Additive notation when the law of internal composition is represented as "a+b{displaystyle a+b}"and the neutral element as "0." On the other hand, the multiplication is the one in which the law of internal composition is represented as "a⋅ ⋅ b{displaystyle acdot b}"or"ab{displaystyle ab}"and the neutral element as "1".

Examples

• (Z,+){displaystyle (mathbb {Z}+)}, the whole set of numbers with the usual sum, is an abelian group; where the neutral element is the 0and the symmetrical xIt is -x.
• (R,+){displaystyle (mathbb {R}+)}, the set of real numbers with the usual sum, is an abelian group; where the neutral element is the 0and the symmetrical xIt is -x.
• (Z {0!,⋅ ⋅ ){displaystyle (mathbb {Z} setminus {0},cdot}}, the set of integers (excluding to 0) with multiplication, is not a group; since the symmetrical element of x That's it. 1/xand said 1/x belongs to the set of rationals, not to the whole (for everything x different from 1 and -1). Note that by not having the zero multipliative symmetrical element, it should be excluded.
• The set of all the bijections of a set X - symbolized by S(X) - along with the composition of functions, is a group (not abelian if the cardinality of X is greater than two) that is called symmetrical group X.
• The set of rectangular matrices with dimensions n× × m{displaystyle ntimes m} with the sum, it's an Abelian group.
• The set of square matrices of order n{displaystyle n} and determinant different from zero with multiplication (general linear group), is not abelian.
• Continuous closed path homotopia classes S1→ → X{displaystyle S^{1to X} based on a certain point, in a topological space XThey form a group not necessarily abelian. This construction is the fundamental group of X.
  • The fundamental group of a circle (S1{displaystyle S^{1}) is the infinite cyclical group; Z{displaystyle mathbb {Z} }.
  • The sphere S2{displaystyle S^{2}} It's trivial. = 0and the same for the n-sphere of higher dimensions.
  • The one with a bull (S1× × S1{displaystyle S^{1}times S^{1}}) is the direct sum Z Z{displaystyle mathbb {Z} oplus mathbb {Z} }.
  • The one with a bull with a removed disc is the free order group two, F2{displaystyle F_{2}}., that of a bull with two deleted discs is F3{displaystyle F_{3}}...
  • The projective plane is Z2{displaystyle mathbb {Z} _{2}}.
  • The one in the Klein bottle has the presentation; a,b:aba=b {displaystyle langle a,b:aba=brangle } and corresponds to the semi-direct product Z{displaystyle mathbb {Z} } with Z{displaystyle mathbb {Z} }.

Morphisms between groups

Between two groups G, H There may be morphisms, i.e. functions that are compatible with operations in each of them. We say an application φ φ :: G→ → H{displaystyle phi colon Gto H} It's a homomorphism if for all pairs of elements a{displaystyle a} and b{displaystyle b} of G{displaystyle G} verified

φ φ (ab)=φ φ (a)φ φ (b){displaystyle phi (ab)=phi (a)phi (b),}

where we used the writing convention ab{displaystyle ab} to indicate the operation a with b in Gand φ φ (a)φ φ (b){displaystyle phi (a)phi (b)} the operation φ φ (a){displaystyle phi (a)} with φ φ (b){displaystyle phi (b)} in H.

If we transform a group switch: aba− − 1b− − 1{displaystyle aba^{-1}b^{-1} is obtained: φ φ (aba− − 1b− − 1)=φ φ (a)φ φ (b)(φ φ (a))− − 1(φ φ (b))− − 1{displaystyle phi (aba^{-1}b^{-1})=phi (a)phi (b)(phi (a))^{-1}(phi (b))^{-1}}}.

Category of groups

From the point of view of category theory, group theory could be categorized as a category called category of groups, because it studies groups and their morphisms. The category of groups is very large, but an equivalence relation can be set up in this category so that it is factored: the relation between groups of being isomorphic reduces structural issues from the category of groups to the category of groups-modulus-the-isomorphs. In this reduction the disjoint union operation converts it into a monoidal category.

Geometric theory of groups

The most current research topics in group theory have to do with modern techniques of topology. A standard way of building new groups from known ones is

• free products and amalgamated free products.
• HNN-extensions.

The great variety of topological techniques can be applied since it is known that it is always possible to construct a topological space (in fact a two-dimensional CW-complex) in such a way that the fundamental group of this space is the given group.

References and notes