Pretty much the same way you show that any set is a subgroup of the group it belongs to. I don't remember what GL2(R) means, but I suspect it consists of 2 x 2 matrices with entries in R, and the operation is probably matrix multiplication.kathrynag said:Homework Statement
Let G=GL2(R)
Show that T=matrix with row 1= a, b and row 2 = 0, d with ad[tex]\neq[/tex]0 is a subgroup of G.
Homework Equations
The Attempt at a Solution
I'm sort of confused on how to show it is a subgroup.
Time to look at the definition...kathrynag said:Ok I guess then I'm kinda confuses on subgroups in general
A subgroup is a subset of a group that shares the same group structure. This means that it contains the identity element, is closed under the group operation, and has inverses for all its elements.
To show that a matrix is a subgroup of a group G, you need to prove that it satisfies the three criteria for a subgroup: it contains the identity element, it is closed under the group operation, and it has inverses for all its elements. You can also show that the matrix has the same group structure as G by demonstrating that it follows the same group axioms.
Yes, a subgroup of a group G can be a proper subset. A proper subset is a subset that is not equal to the original set. This means that a subgroup can have fewer elements than the original group, but it still follows the same group structure.
To prove closure under the group operation for a matrix subgroup, you can take two elements from the subgroup and perform the group operation on them. If the resulting matrix is also in the subgroup, then it is closed under the group operation. This can be shown algebraically by multiplying the two matrices and showing that the result is also in the subgroup.
Showing that a matrix is a subgroup of a group G is important because it allows us to apply the properties and theorems of the group to the subgroup. This can help us simplify calculations and prove results about the subgroup without having to start from scratch. It also allows us to better understand the structure and properties of the original group G.