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From: mward@uoregon.UUCP (mward)
Newsgroups: net.math
Subject: Re: Groups, Alternative Definitions?
Message-ID: <139800005@uoregon.UUCP>
Date: Fri, 7-Mar-86 20:06:00 EST
Article-I.D.: uoregon.139800005
Posted: Fri Mar  7 20:06:00 1986
Date-Received: Wed, 12-Mar-86 21:55:22 EST
References: <370@ihnet.UUCP>
Organization: Univ of Oregon - Eugene, OR
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Nf-ID: #R:ihnet:-37000:uoregon:139800005:000:893
Nf-From: uoregon!mward    Mar  7 17:06:00 1986

Re: Groups, Alternative Definitions?

     Definitions 2 and 4 are not equivalent to the definition
of a group.  Here's a well-known example (see, for instance,
Fraleigh's abstract algebra book):        

      Let S be the set of nonzero real numbers and for
a and b in S define a*b to be |a| times b ( |a| = absolute
value of a ).  This operation is well-defined and associative.
Furthermore, it satisfies Definition 2 (take E = 1 and
Y = |X|^(-1) ).  However, we can readily check that (S,*) is
not a group.  For the only values of E for which E*X = X for
all X in S are E = 1 or E = -1.  Neither value satisfies
X*E = X for all X in S (for E = 1, take any X < 0 and for
E = -1, take any X > 0).  Thus, (S,*) has no identity and is
not a group.

           Michael Ward     UUCP . . .!tektronix!uoregon!mward
           University of    CSNET     mward@uoregon
                     Oregon