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Is the formal logic in these axioms correct?

A group is a nonempty set $G$ with a binary operation $*$ that satisfies the following axioms:

  • $(a * b) \in G \; \forall \;a,b \in G$

  • $a * (b * c) = (a * b) * c \; \forall \; a,b,c \in G$

  • $\exists \; e \in G : a * e = e * a = a \; \forall \; a \in G$

  • $\forall \; a \in G \; \exists \; a^{-1} \in G: a * a^{-1} = a^{-1} * a = e$

In particular, in the last two axioms notice the difference of where the "for all" is placed.
The third axiom is meant to state that there exists an $e$ in $G$ that is a identity for all $a$ in $G$ under $*$.
While the last axiom is meant to state that for every $a$ in $G$ there exists an inverse $a^{-1}$ in $G$ under $*$.

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  • $\begingroup$ correct it is.... $\endgroup$ – tattwamasi amrutam Jun 11 '14 at 5:08
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    $\begingroup$ If you are concerned about the formal logic I think you should be very careful with the placement of quantifiers. I would recommend rewriting the third statement as$$\exists \; e \in G : \; \forall \; a \in G : a * e = e * a = a$$and similarly for the first two. $\endgroup$ – David Jun 11 '14 at 5:13
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    $\begingroup$ The axioms as given are not sentences of first-order predicate calculus. In particular, the placement of the universal quantifiers is not right. There is also nothing connecting the $e$ of the fourth axiom with the $e$ of the fifth. As a very informal description, it is sort of OK. From the formal point of view, quite far from right. $\endgroup$ – André Nicolas Jun 11 '14 at 5:15
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    $\begingroup$ One can avoid the constant symbol by combining the last two sentences suitably into a single sentence. And a single binary function symbol is enough, if that's what we want. A more serious issue for formalization is the $\in$ symbol. In first-order group theory, one would omit all the $\in G$ stuff. $\endgroup$ – André Nicolas Jun 11 '14 at 5:34
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    $\begingroup$ By definition, a binary operation on a set $S$ is a function $S \times S \to S$, so the first axiom is unnecessary. $\endgroup$ – Qiaochu Yuan Jun 11 '14 at 6:01
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You can see List of first-order theories under the heading : Groups.

In addition, you can check the details in every mathematical logic textbook, like Herbert Enderton, A Mathematical Introduction to Logic (2nd - 2001), page 93.

Note

As you can see, there are different possibilities; what make the (little) differences are the symbols used as "basic".

In order to be "formal", you have to specify a language : an alphabet and the syntactical rules for building expressions (see comments above).

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