An alternative definition of a group? Will the following definition of a group work as a basis for group theory:
$\forall G,f,i,e (Group(G,f,i,e)\leftrightarrow f:G\times G\rightarrow G$
$\wedge i:G\rightarrow G$
$\wedge \forall x,y,z\in G (f(f(x,y),z)=f(x,f(y,z))$ (Associativity of $f$)
$\wedge e\in G$
$\wedge\forall x\in G(f(x,e)=x \wedge f(e,x)=x)$ (Identity under $f$)
$\wedge\forall x\in G(f(x,i(x))=e \wedge f(i(x),x)=e))$ (Inverse function $i$)
Edit: This is instead of the usual definition:
$\forall G,f (Group(G,f)\leftrightarrow f:G\times G\rightarrow G$
$\wedge \forall x,y,z\in G (f(f(x,y),z)=f(x,f(y,z))$ 
$\wedge\exists e\in G(\forall x\in G(f(x,e)=x\wedge f(e,x)=x)$
$\wedge\forall x\in G\exists x'\in G (f(x,x')=e \wedge f(x',x)=e)))$ 
 A: Yes, this is the same definition.  The only differences are that your first one doesn't explicitly define the identity and inverse elements to be unique.  However, that can be easily proven  ($x*e^\prime=x\Leftrightarrow x^{-1}xe^\prime=x^{-1}x \Leftrightarrow ee^\prime= e \Leftrightarrow e^\prime= e$, and similarly for the inverses).
A: The two definitions of group are equivalent. Furthermore, the notion of group homomorphism implied by the two definitions turn out to be equivalent as well.
It maybe interesting to contrast the definition of a ring as $(R,+,\times,0,1)$ satisfying equational identities, versus $(R,+,\times)$ satisfying axioms (including the existence of additive and multiplicative units).
While the two definitions define the same objects, they imply different notions of ring homomorphism: by the former definition, a homomorphism $f:R \to S$ should satisfy $f(1_R) = 1_S$. However, this need not be true for the latter definition.
A specific counterexample is $S = \mathbb{Z} / 6$, the integers modulo $6$, and $R = 3S$, the ring whose elements are the elements of $S$ divisible by $3$. The identity map $R \to S$ is a homomorphism according to the notion of homomorphism implied by the second definition of ring, but is not a homomorphism by the one implied by the first definition of ring.
