mikael27 said:Homework Statement
Show that the general linear group GL(3,R) with matrix multiplication is not an abelian
group.
Homework Equations
The Attempt at a Solution
A group to be abelian we have to show that it satisfies the Commutativity.
a*b=b*a
How are we going to show that the general linear group GL(3,R) does not satisfies the Commutativity?
mikael27 said:Take
A=
0 -1 0
1 0 0
0 0 1
B=
1 0 0
0 0 -1
0 1 0
A*B=
0 0 1
1 0 0
0 1 0
B*A=
0 -1 0
0 0 -1
1 0 0
So the two matrices do not commute as A*B=B*A and hence it is not abelian
The General Linear Group is a mathematical concept that represents the set of all invertible matrices of a given order. In other words, it is the group of all possible transformations that can be applied to a vector space without changing its dimension.
A group is considered Abelian if its operation is commutative, meaning that the order in which the elements are multiplied does not affect the end result. In simpler terms, the group's operation is commutative if a*b = b*a for all elements a and b in the group.
The General Linear Group is not Abelian because the operation of matrix multiplication is not commutative. This means that the order in which matrices are multiplied matters, and therefore the group cannot be considered Abelian.
The General Linear Group has various applications in fields such as physics, engineering, and computer science. It is used to represent transformations in quantum mechanics, to solve systems of linear equations in engineering, and to perform operations on images and graphics in computer science.
Yes, there are many other types of groups that are not Abelian. Some examples include the symmetric group, the dihedral group, and the quaternion group. In fact, most groups are not Abelian, making the Abelian property a special and unique characteristic of certain groups.