Lagrangian Notation

Date: 04/08/99 at 21:13:02
From: ben
Subject: Modern Algebra I

We are using Lagrange's Theorem. I think the notation has got me 
flustered and i need some help. How would I find, for example:

   [S4:<(123)>]

Date: 04/09/99 at 11:54:37
From: Doctor Teeple
Subject: Re: Modern Algebra I

Hi Ben,

Thanks for writing Dr. Math.

Lagrange's Theorem says that the order of a subgroup divides the order 
of the group. In case the notation in your example isn't clear, it is 
just asking for the number you get when you divide the order of the 
group by the order of the subgroup. It's called the index of the 
subgroup. (Actually, the index of a subgroup, say H, of a group G is 
defined to be the number of left cosets of H in G, but all you really 
have to do is divide the order of the group by the order of the 
subgroup.) 

So you need to find the order of the groups S_4 and <(123)>. Well, S_4 
is a permutation group with 4 elements, and so it has 4! = 24 
elements. <(123)> is a group of permutations, generated by the element 
(123). To find the other elements in the group, you can just start 
multiplying elements until you get back to the identity. For example, 
since (123) is in S_4, 

   (123)(123) = (132) 

is in S_4, and so on.

So the notation just means:

                     |S_4|        24
   [S_4:<(123)>] = --------- = ---------
                   |<(123)>|   |<(123)>|  

where |G| is the number of elements in group G.

By Lagrange's Theorem, the index is an integer.

I hope that this clears up some of the notation. Please write back if 
it doesn't.

- Doctor Teeple, The Math Forum
  http://mathforum.org/dr.math/