Recent content by seanthinks

Ah I think I got a head of myself, those were merely best and worst case scenarios (if all intersections were nontrivial and trivial, respectively); I keep forgetting that there are multiple ways for these groups to intersect. Alright, I'm not sure I can salvage the argument. Back to the drawing...

Ah, okay. I see what you're saying. Could it be reworked by arguing that a nontrivial intersection of any two Sylow subgroups has exactly order 2? Therefore the elements would have to be of order 2. So the number of distinct elements belonging to the 9 Sylow 2-subgroups is either 9 or 27. But if...

I'm trying to get a little bit more comfortable with using the Sylow theorems and looking back at some examples from my abstract algebra textbook (we used Fraleigh). In an example involving groups of order 36 (the claim is no such groups are simple), the author writes: "Such a group ##G## has...

I'm a math major interested in the field of quantum information science and quantum computing, but I'm unclear on what path to take to explore the field. The field is presented as being at the intersection of physics, computer science, and mathematics but it seems that a majority of the experts...

Thanks for your help!

Actually, scratch that last part, the error in my thinking is glaring to me now lol

Ohhhh okay, I think the problem I'm having is that when I get there I see the solutions as being either ##b = -1/c## or ##c = -1/b##, neither of which seemed particular illuminating. And the solutions 1 and -1, while obvious seemed like I was just blindly guessing rather than reasoning them out...

I got stuck after setting it up as a system of linear equations. I had $$ a^2 + bc = -1 \\ ab + bd = 0 \\ ca + dc = 0 \\ cb + d^2 = -1 $$ which when I solved yielded ##a = -d##, which means ##a^2=d^2##. Then I get stuck on either ##ab + bd## or ##ca + dc## because that means either ##a, b, c...

This is a problem from Lang's Introduction to Linear Algebra. The problem statement is: Find a 2 x 2 matrix A such that A2= ##\begin{pmatrix} -1 & 0 \\ 0 & -1 \\ \end{pmatrix}## = -I The solution is available in the answer section of the book, but it is not shown how the author comes up with...

As indicated, my name is Sean and I've been creeping this forum for like four years now so I figured it's high time I became a member. I've learned a lot from the discussions here, you guys seem like a very fruitful community and I'm excited to be apart of it. Thanks!