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From: lambert@boring.UUCP
Newsgroups: net.math
Subject: Re: Groups, Alternative Definitions?
Message-ID: <6817@boring.UUCP>
Date: Sat, 8-Mar-86 13:26:48 EST
Article-I.D.: boring.6817
Posted: Sat Mar  8 13:26:48 1986
Date-Received: Tue, 11-Mar-86 01:49:31 EST
References: <370@ihnet.UUCP>
Reply-To: lambert@boring.UUCP (Lambert Meertens)
Distribution: net
Organization: CWI, Amsterdam
Lines: 32
Apparently-To: rnews@mcvax

In article <370@ihnet.UUCP> eklhad@ihnet.UUCP writes:
> If we modify the definition of a group slightly, do new groups arise?
> [...]
> There are four possible "weakened" definitions:
>	identity	inverse
> 1.	X*E = X		X*Y = E
> 2.	E*X = X		X*Y = E
> [...]
> If I have not made any serious blunders, each weaker definition implies the
> other three, resurrecting the original concept.

There is an interesting difference between 1, which has the unit and
inverse on the same side of the * operation, and 2, where unit and inverse
are on different sides.  For case 1, it is not necessary to assume
uniqueness of unit and inverse; the existence of at least one unit for the
whole "group" and at least one inverse for each element is sufficient.
Uniqueness can then be proved.

For 2, we only get a group if we assume uniqueness of the unit E.  If we
just assume the existence of *some* unit E, a counterexample is found by
defining X*Y = Y.  This operation is associative: (X*Y)*Z = Z = Y*Z =
X*(Y*Z).  Each element is a left unit.  Take an arbitrary one and call it
E.  Then * satisfies both axioms in 2: E*X = X, and X*Y = E for Y = E.

(All this is well known; at least it was part of the first lecture in an
introductory algebra class I once took.)

-- 

     Lambert Meertens
     ...!{seismo,okstate,garfield,decvax,philabs}!lambert@mcvax.UUCP
     CWI (Centre for Mathematics and Computer Science), Amsterdam