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### 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/
```
Associated Topics:
College Modern Algebra

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