Let $G$ be a group and $V=Var(G)$ be the variety generated by $G$. Suppose the axiomatic rank of $V$ is $n$. Let $Id(V)$ be the set of all identities of $V$.


1- Can we say that every element of $Id(V)$ is equivalent to some identity of the form
$$ w(x\_1, \ldots, x\_n)\approx 1?$$


2- Let $G$ be finite. Can we prove that the axiomatic rank of $V$ is finite without using the fact that $V$ is finitely based?



Two questions about axiomatic rank of groups
Concerning question 2: by a result of Birkhoff, a finite algebra (of finite type) is finitely based if and only if it is of finite axiomatic rank (see e.g. Thm 4.2. in the book of Burris-Sankappanavar). So the question essentially asks for a new proof of the Oates-Powell-Theorem.


