Abelian Group, also known as switching or adding groups, is such a group: . It consists of its own collection G and dual operation*. In addition to satisfying the general group axioms, that is, the combination law of the operation, the unit of G, and the elements of all G are opposite, and also meet the axiom of exchange. Because the group transportation calculation of the Abel group meets the law of exchange and binding, the value of the group elements has nothing to do with the order of the multiplication operation. The concept of Abel group is one of the basic concepts of abstract algebra. The basic research object is molding and vector space. The theory of Abel group is simpler than other non -Abel groups. The limited Abel group has been completely studied. The theory of infinite Abel group is currently being studied. The expansion information: Abele group examples integer and addition operation ” ” is the Abel group, indicating (z, ). The integer, the addition is in line with the combination law, zero is the addition of the law unit, and all integer N has the addition of the method of the addition of the method. Essence All cycle group G is the Abel group. Therefore, the integer set Z forms an Abel group under the Canadian Fa, as is the integer mode. All rings are Abel groups about it. The reversible element in the exchange ring forms the Abel multiplication group. In particular, the real number set is the Abel group under the Canadian Fa. The non -zero -solid collection is the Abel group under the multiplication method. The sub -groups of all Abel groups are regular sub -groups, so each subdler causes business groups. Abel groups, business groups, and direct harmony are also Abel groups. The matrix even is a reversible matrix, which generally does not form an Abel group under the multiplication, because the matrix multiplication is generally unavailable. But some matrix groups are the groups of Abel under the matrix multiplication -an example is the group of 2×2 rotation matrix. Reference information Source: Baidu Encyclopedia-Abel Group
definition 16.10 The operation “*” in the group u003Cg,*> is an exchanging operation, which is called the group u003Cg,*> is an exchange group ( Group) (Abel group).
example 16.18 group u003Cz, >, u003Cr, >, u003Cq, >, u003Cc, > all exchanges group.
Theorem 16.16 Set u003Cg,*> is a group, then u003Cg,*> The sufficient necessary conditions for the exchange group are:
There are (a*b) 2 = a2*b2
proved the necessity: for A, B∈g, because the operation “*” is exchanged, so there is
(a*b) 2 = (a*b)*(a*b) = a*(b*a)*b
= a*(a*b)*b = (a* a)*(b*b) = a2*b2
P everyone and sufficient: for A, B∈g, if (a*b) 2 = A2*b2, then
(a*b)*(a*b) = (a*a)*(b*b)
a*(b*a)*b = a*(a*b)* B
This to know: B*a = a*b
So the operation “*” meets the exchange law, that is, group u003Cg,*> is the exchange group.
Pay content for time limit to check for freenAnswer Hello, there are still differences between WeChat and QQ, but they are almost the same. WeChat joins the exchange group, as long as someone is in the group, you can pull you into it. QQ joining the exchange group can refer to the following methods. The specific operation steps are as follows: 1. Click the “Find” button in the menu bar below the QQ; 2. Click “Finding the Group” to enter the “exchange” related name in the search; 3. According to your own seek Group, click “Add Group”; 4. Wait for the administrator to verify; the group members change the group, you can send a message in the group to change the group, or you can directly find someone to talk about the group. But slowly it will gain.
Abelian Group, also known as switching or adding groups, is such a group:
. It consists of its own collection G and dual operation*. In addition to satisfying the general group axioms, that is, the combination law of the operation, the unit of G, and the elements of all G are opposite, and also meet the axiom of exchange. Because the group transportation calculation of the Abel group meets the law of exchange and binding, the value of the group elements has nothing to do with the order of the multiplication operation.
The concept of Abel group is one of the basic concepts of abstract algebra. The basic research object is molding and vector space. The theory of Abel group is simpler than other non -Abel groups. The limited Abel group has been completely studied. The theory of infinite Abel group is currently being studied.
The expansion information:
Abele group examples
integer and addition operation ” ” is the Abel group, indicating (z, ). The integer, the addition is in line with the combination law, zero is the addition of the law unit, and all integer N has the addition of the method of the addition of the method. Essence All cycle group G is the Abel group. Therefore, the integer set Z forms an Abel group under the Canadian Fa, as is the integer mode.
All rings are Abel groups about it. The reversible element in the exchange ring forms the Abel multiplication group. In particular, the real number set is the Abel group under the Canadian Fa. The non -zero -solid collection is the Abel group under the multiplication method. The sub -groups of all Abel groups are regular sub -groups, so each subdler causes business groups. Abel groups, business groups, and direct harmony are also Abel groups.
The matrix even is a reversible matrix, which generally does not form an Abel group under the multiplication, because the matrix multiplication is generally unavailable. But some matrix groups are the groups of Abel under the matrix multiplication -an example is the group of 2×2 rotation matrix.
Reference information Source: Baidu Encyclopedia-Abel Group
Exchange group (Abel group)
definition 16.10 The operation “*” in the group u003Cg,*> is an exchanging operation, which is called the group u003Cg,*> is an exchange group ( Group) (Abel group).
example 16.18 group u003Cz, >, u003Cr, >, u003Cq, >, u003Cc, > all exchanges group.
Theorem 16.16 Set u003Cg,*> is a group, then u003Cg,*> The sufficient necessary conditions for the exchange group are:
There are (a*b) 2 = a2*b2
proved the necessity: for A, B∈g, because the operation “*” is exchanged, so there is
(a*b) 2 = (a*b)*(a*b) = a*(b*a)*b
= a*(a*b)*b = (a* a)*(b*b) = a2*b2
P everyone and sufficient: for A, B∈g, if (a*b) 2 = A2*b2, then
(a*b)*(a*b) = (a*a)*(b*b)
a*(b*a)*b = a*(a*b)* B
This to know: B*a = a*b
So the operation “*” meets the exchange law, that is, group u003Cg,*> is the exchange group.
Pay content for time limit to check for freenAnswer Hello, there are still differences between WeChat and QQ, but they are almost the same. WeChat joins the exchange group, as long as someone is in the group, you can pull you into it. QQ joining the exchange group can refer to the following methods. The specific operation steps are as follows: 1. Click the “Find” button in the menu bar below the QQ; 2. Click “Finding the Group” to enter the “exchange” related name in the search; 3. According to your own seek Group, click “Add Group”; 4. Wait for the administrator to verify; the group members change the group, you can send a message in the group to change the group, or you can directly find someone to talk about the group. But slowly it will gain.