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I am working on some introductory notes for group theory. Comments on my initial approach here and any errors so far would be appreciated.

I begin with the group axioms:

  1. $\forall a,b\in G:[ a+b\in G]$

  2. $\forall a,b,c\in G:[ (a+b)+c=a+(b+c)]$

  3. $0\in G$

  4. $\forall a\in G: [a+0=a \land 0+a=a]$

  5. $\forall a\in G: \exists b\in G: [a+b=0 \land b+a=0]$

Then I prove:

  1. $\forall a,b \in G:[a+b=a \implies b=0]$ (All right identities are equal to $0$)

  2. $\forall a,b \in G:[a+b=b \implies a=0]$ (All left identities are equal to $0$)

  3. $\forall a,b,c\in G:[a+b=c+b \implies a=c]$ (+ is right-cancelable)

  4. $\forall a,b,c\in G:[a+b=a+c \implies b=c]$ (+ is left-cancelable)

  5. $\forall a,b,c\in G:[a+b=0 \land a+c=0 \implies b=c]$ (right inverses wrt $0$ are unique)

  6. $\forall a,b,c\in G:[a+b=0 \land c+b=0 \implies a=c]$ (left inverses wrt $0$ are unique)

12.$\exists !inv: \forall a\in G: [inv(a)\in G\land a+inv(a)=0 \land inv(a)+a=0]$ (unique inverse operator exists)

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    $\begingroup$ Motivation is very important. I would suggest including a lot of it. Also, group operations are traditionally written multiplicatively rather than additively if they are not assumed abelian. $\endgroup$ – Michael Joyce Apr 5 '16 at 21:11
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    $\begingroup$ I'd second @MichaelJoyce's implied recommendations to give examples and motivation, as well as making clear that non-abelian groups can be considered, and there the operation is written multiplicatively. Although succinctness has a terrific appeal as a summary after the fact, I think people learning things benefit from a more discursive introduction to new things, with examples. Without examples, how do we know what phenomena we're trying to "fit"? Terse summary of axioms afterward. $\endgroup$ – paul garrett Apr 5 '16 at 22:30
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    $\begingroup$ @DanChristensen It is worth mentioning the context you are teaching this in. I assume (from looking at your user profile a bit) that you do not have a class of students, but rather these examples would exist to showcase your computer symbolic logic system? In that case everything seems fine except for (as noted above) mathematicians almost never write $+$ for a noncommutative operation. Your axiom 1 is also usually not needed: either the operation is defined as a function $G \times G \to G$, in which case this is automatic, or the axioms are formulated in FOL. $\endgroup$ – Steven Gubkin Apr 6 '16 at 14:39
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    $\begingroup$ I also somewhat question making basic group theory a showcase of your system if you are not very well versed in the subject. If you want to learn it, probably best to learn it thoroughly first. $\endgroup$ – Steven Gubkin Apr 6 '16 at 14:41
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    $\begingroup$ Are these notes for teaching group theory? Or notes for self-study? Who is the audience? Will they be distributed or just used for planning? $\endgroup$ – mweiss Apr 6 '16 at 16:29
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My comment is: what are you aiming to achieve with these notes? My instinctive reaction is that you are doing too much and not getting the student to do enough. Writing out a full exposition is fine if the notes are for reference or you only care about memorisation, but won't help that much if the aim is to learn to do mathematics.

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  • $\begingroup$ These notes are meant as a resource for instructors and a supplement to textbooks and lectures on group theory and the general methods of rigorous mathematical proof for undergrads. Students should know what a truly formal proof looks like. $\endgroup$ – Dan Christensen Apr 10 '16 at 21:07

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