Find Order of Elements & Matrices: A Comprehensive Guide

[dlin3]

I'm quite confused what order of elements consists of. I understand that the order of an element g in a group G is the smallest positive integer n such that g^n = e. And I also understand that to find the order of a group element g, you compute the sequence of products g, g^2, g^3,… until you reach the identity for the first time. The exponenet of this product is the order of g. If the identity never appears in the sequence, then g has infinite order.

For example,
For U(15) = {1,2,4,7,8,11,13,14} under multiplication modulo 15. This group has order 8. To find the order of the element 7, say, we compute the sequence 7^1= 7, 7^2= 4, 7^3= 13, 7^4= 1, so |7|=4. But, how do you find that 1,2,4,7,8,11,13,14 are part of U(15)? if one did not give you the set, how would you be able to find the order of U(15)? or even the order of 7 (|7|)?

A Bigger problem I have,
How do you find order for matrices?
like if A= [0, -1, 1, 0] what is the order of |A|?

Please help! I am so confused on how to find order! I reallly appreciate it! Thanks!

 

[Tom86]

You've got the right idea about group elements of finite order, though you say to compute the order of an element you examine the sequence g, g^2, g^3 etc. etc. until you return to the identity element. This is not wrong, however there is a result in group theory that will almost always be proved fairly early on in a first abstract algebra course called Lagrange's Theorem which states that the order of any subgroup H of a finite group G divides the order of G (note I'm talking about the order of a group, not an element). An immediate consequence of this (via a little bit of cyclic subgroup theory) is that the order of any element must divide the order of the group itself. So for example if you have a group of order 8 (e.g. U(15)), the elements must have orders of powers of 2, since 8=2^3. So instead of computing g, g^2, g^3, etc. you only need to compute g, g^2, g^4 and g^8.

That was a bit long-winded (and possibly not very clear) and I'm sorry if it's a result you already know!

To your other questions - the group of units U(n) of congruence classes modulo n is usually covered in a first or second course on elementary number theory. Notice in U(15) that every "integer" (I've put integer in inverted commas because in this context we are technically dealing with congruence classes) is relatively prime to 15. This is not a coincidence - the group structure of U(n) is made up of all "integers" (congruence classes) relatively prime to n. Thus the order of U(n) follows immediately and is given by phi(n) where phi is the Euler phi (or totient) function.

The order of a matrix is potentially a more complicated problem. Some have finite order, and others don't. I'm afraid to go to far into this topic for fear of not knowing enough about it - I've only recently graduated university and the one course I had in group theory was woefully inadequate. I spent most of my final year concentrating on ring theory/galois theory/algebraic number theory and never really did more than I needed to as far as group theory is concerned. I'm about to start a Master's course with modules in advanced group theory and representation theory, which I'm really looking forward to!

I hope at least some of this has been intelligible/helpful!

 

[NoMoreExams]

Think about what order is, it's the number of times you have to perform the operation (whatever the operation is for YOUR group) so that you get back the identity (whatever the identity in YOUR group is). So for your matrix, you haven't defined the operation nor the identity, kind of hard to answer that question.