A group is a simple algebraic structure with only one operation;It can be used to build many otherAlgebraic systemA basic structure of.
Let G be a group and H be its nonempty subset.If H is closed to group multiplication, that is, h ∈ H, h, k ∈ H, then the algebraic operation on G naturally induces an algebraic operation on H.If H forms a group under this algebraic operation, then H is called a subgroup of G.[4]
Subgroups aregroupA special nonempty subset of.The nonempty subset H of group G, if the multiplication of G also becomes a group, then H is called a subgroup of G and recorded as H ≤ G.If the subgroup H ≠ G, then H is called a proper subgroup of G, denoted H G or simply denoted HI | i ∈ I} is the set of subgroups of G, I is an index set, then all HiDelivery of HiIs a subgroup of G.[1]
The concept of group
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A relatively simple algebraic structure with only one operation;It can be used to build many otherAlgebraic systemA basic structure of.
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.
The group satisfying the commutative law is calledCommutative 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,LagrangeWhen discussing the permutation between the roots of algebraic equations, we first introduce the concept of group, whose name isGalois It was first proposed in 1830.
Discrimination of subgroups
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There are the following propositions about the discrimination of subgroups of a group:
1. Let H be a nonempty subset of a group, then H is a subgroup of G if and only if H satisfies one of the following two conditions:
(1) For any a, b ∈ H, a · b ∈ H and a ^ (- 1) ∈ H;
(2) For any a, b ∈ H, a · b ^ (- 1) ∈ H.
Any group has two trivial subgroups: G and e, where e is G'sUnitary。[2]
Basic properties of subgroups
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H is a subgroup of group G if and only if it is a nonempty set and closed under the product sum inverse operation.(Closed condition means that any two elements a and b, ab and a − 1 in H are in H.These two conditions can be combined into an equivalent condition: any two a and b in H, ab − 1 will also be in H.)If H is finite, then H is a subgroup if and only if H is closed under the product.(In this case, each element a of H will generate a finite cyclic subgroup of H, and the inverse element of a will be a − 1=an − 1, where n is the order of a.)
The above conditions can be described by homomorphism;That is, H is a subgroup of the group G if and only if H is a subset of G and there is an endomorphism mapped from H to G (that is, for each a, i (a)=a).
The identity element of a subgroup is also the identity element of a group: if G is a group with the identity element eG, and H is a subgroup of G with the identity element eH, then eH=eG.
The inverse element of an element in a subgroup is the inverse element of this element in the group: if H is a subgroup of group G, and a and b are elements in H that will make ab=ba=eH, then ab=ba=eG.
The intersection of subgroups A and B is also a subgroup.But its union is also a subgroup if and only if A or B contains another one, for example, 2 and 3 are in the union set of 2Z and 3Z, but the sum of 5 is not.
If S is a subset of G, then there is a minimal subgroup including S, which can be found by obtaining the intersection of all subgroups including S;This minimal subgroup is marked and called the subgroup generated by S.An element in G is inner if and only if it is a finite product of elements in S and its inverse.
Each element a in group G will generate a cyclic subgroup。ifIf Z/nZ is isomorphic to a positive integer n, then n will be the smallest positive integer that will make an=e, and n is called the "objective" of a.ifIf it is isomorphic to Z, then a will be called "infinite eye".
The subgroups of any given group will form a complete lattice under inclusion, which is calledSubgroup lattice。(The maximum lower bound is the intersection of general set theory, and the minimum upper bound of a group of subgroups is the subgroup "generated" by the set theory union of these subgroups.) If e is the unit element of G, then of course its group {e} will be the smallest subgroup of the group G, and its maximum subgroup will be the group G itself.[3]
