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From: eklhad@ihnet.UUCP (K. A. Dahlke)
Newsgroups: net.math
Subject: Groups, Alternative Definitions?
Message-ID: <370@ihnet.UUCP>
Date: Tue, 4-Mar-86 09:37:40 EST
Article-I.D.: ihnet.370
Posted: Tue Mar  4 09:37:40 1986
Date-Received: Wed, 5-Mar-86 05:29:15 EST
Distribution: net
Organization: AT&T Bell Laboratories
Lines: 46

If we modify the definition of a group slightly, do new groups arise?

I have never had a group theory course, so if this is all standard established
knowledge, forgive me.  I thought the question was interesting.

Review:	A group is a set of elements S and an operator '*',
possessing the following properties:
The binary operator '*' is a well defined function, closed in S.
The operator '*' is associative.
There is a unique identity element E, such that X*E = E*X = X for every X.
Every X has a unique inverse Y, such that X*Y = Y*X = E.

Consider the identity and inverse properties of a group, as outlined above.
Suppose the order of the operands is significant.
Example: every X has a unique inverse Y, such that X*Y = E.
No constraints placed upon Y*X.
There are four possible "weakened" definitions:
	identity	inverse
1.	X*E = X		X*Y = E
2.	E*X = X		X*Y = E
3.	E*X = X		Y*X = E
4.	X*E = X		Y*X = E
If I have not made any serious blunders,
each weaker definition implies the other three,
resurrecting the original concept.

By symmetry, definitions 1 and 3 are equivalent, producing reflected theorems.
Similarly, 2 and 4 are equivalent.  Therefore, we consider only 1 and 2.

Suppose we adopt definition 1.
Every X has an inverse Y, and this Y must have an inverse Z.
Multiply the equation X*Y = E by Y on the left and Z on the right.
After invoking associativity and identity properties,
we have Y*X*(Y*Z) = Y*Z.
Since Z is the inverse of Y, this reduces to Y*X = E.
Thus, if Y is the inverse of X, then X is the inverse of Y.
To prove E*X = X, begin with Y*E = Y, where Y is the inverse of X.
Multiply by X on the right and on the left, and reduce the equation to E*X = X.
Thus, definition 1 implies all the original axioms.

Starting with definition 2 is more challenging.
I will leave this as an exercise for the interested.

-- 
	When the sky becomes uranious, we will all go simultaneous.
			Karl Dahlke    ihnp4!ihnet!eklhad