.So I'm taking abstract algebra, and we are using Artin's book Algebra. I'm trying to catch up over spring break after falling horribly behind, and I was hoping for a little illumination on some of the hw problems.
A) Is the set of positive integers in Z+ a subgroup? (I assume Z+ means the set of integers with operation addition.)
So I want to test 3 things:
I) if a,b in subgroup H, then ab in H -- yes, because any two positive integers added together will yield another positive integer
II) identity in H -- yes, because 0 + a = a so 0 is the identity which is in H
Hmmm... is zero (0) a POSITIVE integer?
III) if a in H then inverse of a in H -- no, because the only element in Z that when added with a positive integer yields 0 is a negative integer.
B) Is the set of 2x2 matrices with the only non zero entry in the (1,1) place (upper left) a subset of GL2(R)?
I don't see how it can be in GL2(R), because it's determinant will always be 0 and the determinant of all matrices in GL2(R) is always non-zero.
As simple as that......and thus the answer is: no, it is not a subset of all the 2x2 invertible real matrices.
If someone could let me know if I'm heading in the right direction, or if not ask some questions that will get me thinking about why not (I at least know enough not to expect a straight answer if I'm wrong).