An obsession?
Since I’d like to write about number theory, I can’t an urge to introduce the ring of integers from scratch. Thus first I am about to post about Abelian groups. Since I am about to post about Abelian groups, I jusrt have to introduce groups in general. This my unrealistic obsession with systematic writing is my weakness. I don’t know the secret of pragmatic writing from the middle, which is an art mixed with self-assurance or arrogance, depending on the author.
I’ve already wrote about axioms of groups, together with the first simple theorems, in the past at least twice. Once on Chimeryd site, a long time ago. That site unfortunately has dissappeared. Thus a couple years ago I wrote about groups again. That text must be still somewhere on the Internet, in one of my sites/pages, but I was unable to find it. Ok, I am not happy about it but I’ll still write about groups for the third time (the last? :)).
It’s amazing that three simple and natural axioms have at the same time so many interesting consequences (some of them deep), and they define a theory–a true profound theory, not just a loosely defined theme– which admits a whole world of models (examples), from finite to infinite, from discrete to analytic, from easy to understand to most complex.
The importance of the groups comes from their actions on other mathematical objects, thus throwing light on the structure of these objects, as well as on the group itself (we study groups by studying their actions). Thus it is good to remember about the group transformation spirit of groups. Even when studying a group alone by itself it’s good to view its elements as acting at least on itself.
A quick intro to groups (a sketch).
Let me write here a preliminary sketch, before I’ll post a more finished version on the main part of wlod.net.
Axioms
DEFINITION A group is an ordered quadruple
Γ := (G * Inv e)
where G is a set, e ∈ G, * and Inv are a unary and binary operation in  G, such that the following axioms hold:
- (x*y)*z = x*(y*z)
- e*x = x
- Inv(x)*x = e
It’s good to interpret these axioms in terms of the action of the group on itself, meaning that with every element a ∈ X we associate a (left) map
La : G → G
given by:
∀ x ∈ G La(x) := a*x
First goal
Axioms 2 3 above allow symmetric versions:
- x*e = x
- x*Inv(x) = e
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I will show below that these symmetric variations are theorems, that they follow from the axioms 1-3 of the theory group. It’s an elementary result (obviously :-)) but not trivial. (It goes without saying that this topic is classical; I don’t claim any of the results, of course not, and only the exposition is mine, at least to my limited knowledge).
Simple general properties
THEOREM 0 e*e = e
PROOF substitute x := e in axiom 2 above.
THEOREM 1 If a*x = a*y then x=y, for arbitrary a x y ∈ X. In other words, transformation La is an injection of X into itself.
PROOF If a*x = a*y then:
x = Inv(a)*a*x = Inv(a)*a*y = y
END of PROOF
REMARK More explicitly, equation a*x = b has at the most one solution, namely
x :=Inv(a)*b
But is this x actually a solution? At this moment this is still an open question, which will be answered in positive soon.
THEOREM 2 ∀x ∈ X x*e = x
PROOF
- Inv(x)*(x*e) = (Inv(x)*x)*e = e*e = e
- Inv(x)*x = e
Thus, by theorem 1, x*e = x. END of PROOF
THEOREM 3 ∀x ∈ X x*Inv(x) = e
PROOF By Theorem 2:
Inv(x)*e = Inv(x) = e*Inv(x)
= (Inv(x)*x)*Inv(x) = Inv(x)*(x*Inv(x))
and by Theorem 1, x*Inv(x) = e. END of PROOF