the group of invertible 2 by 2 matrices

[yxgao]

This is my third day studying number theory and abstract algebra, and I'm confused about the concept of subgroups.

Which one or more of the following are subgroups of GL(2,R), the group of invertible 2 by 2 matrices with real entries under matrix multiplication?

I. T = {All A in GL(2,R) for which det A = 2}
II. U = {All A in GL(2,R) for which A is upper triangular.}
III. V = {All A in GL(2,R) for which trace(A) = 0}

The answer is II only.

My textbook has an explanation, but does not go into too much detail.

What are ALL the reasons for which the I. and III. are wrong and II. is correct? E.g., which subgroups are closed under multiplication, have the identity, and have an inverse?

[NateTG]

Actually, it doesn't make much sense to ask about inverses when there is no identity.

That said, if you look at I, you will discover that I is not closed over multiplication.

For III, you can see that the trace of the identity matrix is non-zero, so there is no identity, even if the set is closed under multiplication.

[selfAdjoint]

What determinant and trace does the unit matrix in GL(2,R) have?

What are the axioms for a group?