Torsion-free simple linear group

In summary, the speaker is considering choosing the topic of the existence of torsion-free simple linear groups for their undergraduate math thesis, but they are unsure if it is recommended. They have found a resource for examples, but it is difficult to find a suitable one. The speaker mentions books by Wehrfritz, Humphreys, Shafarevich-Kostrikin, and Springer as important resources to check. They also mention a professor's warning about the difficulty of the topic.
  • #1
kimkibun
30
1
im having my undergrad math thesis right now, and the topic that i want to choose is the existence of torsion-free simple linear groups. (which i found here: http:///www.sci.ccny.cuny.edu/~shpil/gworld/problems/probmat.html)

i just want to know if this topic is recommended for undergrad?
 
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  • #2
kimkibun said:
im having my undergrad math thesis right now, and the topic that i want to choose is the existence of torsion-free simple linear groups. (which i found here: http:///www.sci.ccny.cuny.edu/~shpil/gworld/problems/probmat.html)

i just want to know if this topic is recommended for undergrad?



The link is broken\doesn't exist. Check it, please

About the subject: it sounds interesting but difficult. Right now I can't think of one single example that fulfills all the conditions, so I guess any such must be a rather vicious one.

Anyway, the books by Wehrfritz, Humphreys, Shafarevich-Kostrikin or Springer are important to check. In particular, and since you may also want to consider topological groups and not merely abstract, discrete ones, Humphreys makes a nice observation: an almost simple algebraic group G (i.e., without closed connected normal subgroups except the trivial ones) is such that G/Z(G) is ALWAYS simple in the usual, abstract sense.

DonAntonio
 
  • #3
DonAntonio said:
The link is broken\doesn't exist. Check it, please

here's the link sir

http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/probmat.html

i agree with you sir when it comes to difficulty. my professor said that its a suicide for a graduating student to choose such topic. well anyway, thank you sir for your reply. btw, i enjoy reading the book of Shafarevich-Kostrikin.
 

1. What is a torsion-free simple linear group?

A torsion-free simple linear group is a type of mathematical group that consists of matrices with real or complex entries and certain properties. These groups are important in the study of linear algebra and have applications in many areas of mathematics and physics.

2. Can you give an example of a torsion-free simple linear group?

Yes, the group of invertible n x n matrices with real or complex entries, denoted by GL(n,R) or GL(n,C), is a torsion-free simple linear group.

3. What does it mean for a linear group to be torsion-free?

A linear group is torsion-free if it does not contain any non-trivial elements with finite order. In other words, there are no elements in the group that, when multiplied by itself a certain number of times, will result in the identity element.

4. How is a simple linear group different from a general linear group?

A simple linear group is a linear group that does not have any non-trivial normal subgroups, meaning that it cannot be broken down into smaller groups. On the other hand, a general linear group can have non-trivial normal subgroups.

5. What are the applications of torsion-free simple linear groups?

Torsion-free simple linear groups have various applications in pure mathematics, including group theory, representation theory, and algebraic geometry. They also have practical applications in physics, particularly in the study of symmetry and geometric structures.

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