Let E and F be twoGroupoid, twoMonoid, two groups, two rings, two vector spaces, twoAlgebraOr two unitary algebras.The mapping f from E to F is said to be isomorphic if f has an inverse mapping and f and f-1 are two homomorphisms.
Let E and F be two ordered sets.The mapping f from E to F is isomorphic if it has an inverse mapping and f and f-1 are increasing.That is, for any element pair (x, y) of E, the relationship x ≤ y is equivalent to f (x) ≤ f (y).If E and F are totally ordered sets, we can prove that any bishomomorphism is isomorphic.For example, the exponential function x ↦ ex is isomorphic from the real number addition group R to the strict positive real number multiplication group R *+.The inverse isomorphism is the logarithmic function x ↦ lnx.Both are increasing, and these two bijections are also isomorphic to ordered sets.[1]
Two mathematical systems (such as two algebraic systems) are isomorphic when their elements and the operations defined by them correspond to each other one by one, and the results of the operations also keep one-to-one correspondence, which is denoted as Γ.They have the same structure for the operations defined.For example, decimal numbers are isomorphic to binary numbers.
The mapping of establishing isomorphic relations is called isomorphic mapping.For example, when the mapping is one-to-one mapping and the corresponding elements keep corresponding with respect to the operation, it is isomorphic mapping.
Isomorphism is one of the most important concepts in mathematics.In many cases, a difficult problem can often be transformed into another isomorphic, seemingly unrelated, solved problem, so that the original problem can be easily solved.Although mathematics has become more and more complex, the use of isomorphic concepts not only simplifies mathematics, but also makes mathematics more and more unified.The results that appear different on the surface but are essentially equivalent can be expressed in a unified form.For example, if the four-color theorem is proved, dozens of assumptions isomorphic to it in other mathematical branches are also proved.
Let G and G 'be two groups, if there is a | - | mapping σ from G to G', so that longitude (ab)=σ (a) σ (b) holds for all a, b ∈ G, then G is isomorphic to G ', recorded as G ≅ G'.The mapping suitable for the above conditions is called isomorphism mapping (or isomorphism for short).The isomorphism of group G to itself is called automorphism.
Automorphism
Announce
edit
Let E be a groupoid, monoid, group, ring, vector space, algebra or unitary algebra.The isomorphism from E to itself is called the automorphism of E.
After the composition rule (f, g) ↦ g ° f is given, the automorphism set of E is a group, which is naturally called the automorphism group of E, and is recorded as Aut (E). For example, let E be a vector space on the commutative body KThe isomorphic similarity of E is automorphism if and only if its ratio is nonzero——Let's assume that E is finite dimensional.In order for the endomorphism of E to be automorphism, it must be monomorphic or bimorphic.[2]
group
Announce
edit
A group is a simple algebraic structure with only one operation;It is a basic structure that can be used to establish many other algebraic systems.
Let G be a nonempty set, and a, b, c be its arbitrary elements.If an algebraic operation "·" (called "multiplication" and the operation result called "product") defined for G satisfies:
(1) Closure, a · b ∈ G;
(2) Associative law, i.e. (a · b) c=a · (b · c);
(3) For any element a, b in G, there are unique elements x, y in G, so that a · x=b, y · a=b, then G is said to form a group for the defined operation "·".For example, all real numbers that are not equal to zero form a group with respect to the usual multiplication;Turn clockwise (about modulo 12 addition) to form a group.
A group satisfying the commutative law is called a commutative group.
Group is one of the most important concepts in mathematics, which has penetrated into all branches of modern mathematics and other disciplines.Whenever symmetry is involved, there is a group.For example, various geometries can be defined by studying the properties of graphs that remain unchanged under the transformation group, that is, the transformation group can be used to classify geometries.It can be said that without understanding groups, it is impossible to understand modern mathematics.
In 1770, when Lagrange discussed the replacement between the roots of algebraic equations, he first introduced the concept of group, whose name was first proposed by Galois in 1830.
Automorphism group
Announce
edit
A special group.It refers to the group formed by the mapping of the group itself.A group formed by all automorphisms of group G under the composition operation of mapping is called the automorphisms group of group G, which is often recorded as Aut (G).
Important geometric transformation group.It is the basis of geometric classification.Let S be a given space, and U be a graph on S. If a transformation g from S to itself changes U to U itself, then g is called an automorphism transformation on U, or automorphism on U for short.All the automorphisms of U on S form a transformation group, which is called automorphisms of U.The shape U that remains unchanged in the transformation is called absolute shape.For example, if a straight line is taken on the projective plane as an infinite straight line, the automorphism projective transformation that keeps the infinite straight line unchanged on the projective plane forms a transformation group. It is an automorphism group about the infinite straight line, and it is also two-dimensionalProjective transformation groupThe subgroup of, that isAffine transformation group。
Automorphism group of graphs
Announce
edit
Also called node cluster.An important group in graph theory.It is a group formed by all automorphisms of graph G.A graph whose automorphism group is a trivial group is called a monograph.Ko ¨ nig, D. proved that any finite abstract group is isomorphic to the automorphism group of a graph.All endomorphisms of graph G form a semigroup, which is called the endomorphism semigroup of graph G.Any finite semigroup with identity element is isomorphic to the endomorphism semigroup of a graph.The automorphism group acting on the edge set E of graph G is called the edge group of G, also called the line group.The group of a composite graph obtained from the operation of various graphs is represented by the composition of the groups of the graphs that constitute it, which is called the group of a composite graph.The following results are commonly used:[3]
1. Group of complementary graph: Γ (G -)=Γ (G)
2. Groups of isomorphic disjoint union graphs: Γ (nG)=Sn[Γ(G)].
3. Groups of disjoint union graphs with different constructions:
