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### What is a Loop?

```
Date: 10/06/97 at 11:11:04
From: Stephen Gardner
Subject: Loops

This may be a hideously simple question (which will probably make it
easy to answer), but what is a loop? Further, exactly what is _an_
algebra? Thirdly what is a Lie group?
```

```
Date: 10/06/97 at 15:37:56
From: Doctor Rob
Subject: Re: Loops

A loop is a set with one operation which is:
1. closed under the operation,
2. has a two-sided identity,
3. has inverses.

A loop with the Associative property would be a group. There is a
really neat example of a loop of order five with every non-identity
element of order two.  Here is a multiplication table:

x | 1 a b c d
-------------
1 | 1 a b c d
a | a 1 d b c
b | b c 1 d a
c | c d a 1 b
d | d b c a 1

Loops can be commutative or not.

An algebra can be thought of as a ring with a vector space structure
over a field. It can also be thought of as a vector space over a field
on which you define multiplication of vectors. An example is the ring
of n-by-n matrices over a field. You can add them or multiply them by
a scalar, and you get the vector space parts of the definition. You
can also multiply them by each other, and that gives you the ring
parts of the definition.

A Lie group is a set G such that:
1. G is a group;
2. G is a paracompact, real analytic manifold; and
3. The mapping from G x G --> G defined by (x,y) --> xy^(-1) is real
analytic.

I don't like this definition much, since I am not a topologist or an
analyst. The simplest example I have found is as follows. Let V be a
finite-dimensional real vector space, and L(V) the associative algebra
of linear endomorphisms of V. The general linear group GL(V)={x in
L(V):det(x)<>0} is a Lie group. If you pick a basis of n elements for
V, then L(V) are the linear transformations from V to V, and GL(V) is
the group of nonsingular n-by-n real matrices, GL(n,R). The topology
is the induced topology gotten from the ordinary topology on the real
numbers.

I'm not sure the last part is very satisfying, but it is a complicated
concept. It was invented to study the relationship between
differential equations and the groups of operators which preserve
their solutions.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Linear Algebra
High School Sets

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