eklhad@ihnet.UUCP (K. A. Dahlke) (03/04/86)
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
ladkin@kestrel.ARPA (Peter Ladkin) (03/08/86)
In article <370@ihnet.UUCP>, eklhad@ihnet.UUCP (K. A. Dahlke) writes: > If we modify the definition of a group slightly, do new groups arise? You might be interested in loops, where the multiplication has different left and right inverses. Unfortunately, elementary accounts are rare. There is a volume in the Springer Ergebnisse series on loops, written by Bruck. They're useful for representing Latin Squares. There's a more recent reference which currently escapes me. Peter Ladkin
mward@uoregon.UUCP (mward) (03/08/86)
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
Oregonlambert@boring.UUCP (03/08/86)
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
gwyn@brl-smoke.UUCP (03/09/86)
Karl Dahlke observes that the group axioms need only specify right- (or left-) inverses instead of two-sided inverses, since the latter may be deduced from the former. There are other possible formulations of the group axioms. For example, uniqueness of inverse can be dropped since it follows as a consequence of the weaker formulation anyway. Textbook presentations of group theory (e.g. Herstein) usually have several exercises in which one is to show that an apparently weaker set of axioms for a (set,operator) is sufficient to make the (set,operator) constitute a group. Group theory news item: The latest BAMS contains a tutorial article by Gorenstein on the group classification theorem.
mward@uoregon.UUCP (mward) (03/10/86)
In my earlier response to Karl Dahlke's alternative
formulations of the group axioms I carelessly overlooked
the word "unique" in his definitions. The example I
mentioned shows that without the assumption of the uniqueness
of the left identity, the existence of a left identity and right
inverses does not always yield a group. In the presence of
the uniqueness assumption on the left identity, a group is
obtained.
Michael Ward UUCP . . .!tektronix!uoregon!mward
CSNET mward@uoregon