I apologize if this question is meaningless or trivial:


What are examples of **Algebras** admitting quantifier elimination? Especially are there **Groups** admitting quantifier elimination?


I need to say some words about my motivation: last week I was proved some results concerning relatively free algebras in varieties, as an example the following result was proved:


**Theorem** (false) Let $\mathbf{V}$ be a variety of algebras and $X$ be a set. Then the relatively free algebra $F\_{\mathbf{V}}(X)$ is $u\_{\omega}$-compact. 


A communication with Anton Klaychko revealed to me that my argument had mistake. The mistake was funny: I was replaced $\forall x \varphi(x) \Leftrightarrow \forall x \psi(x)$ by $\forall x( \varphi(x)\Leftrightarrow \psi(x))$ implicitly in my proof. Today, I learned some elementary facts about quantifier elimination and assuming that $\mathbf{V}=Var(A)$ with $A$ an algebra admitting quantifier elimination, I modified my proof. Now I have a correct proof of the above theorem with the extra assumption:


1- $\mathbf{V}=Var(A)$,


2- $A$ admits quantifier elimination.


I need some examples of such algebras to complete my work. Before I learned about some well-known examples such as algebraically closed fields and some Boolean algebras (and some other examples which are not algebras and so are not relevant). 



Algebras admitting quantifier elimination
The finite groups with quantifier elimination are classified in 


Cherlin, Gregory; Felgner, Ulrich, Homogeneous finite groups. 
J. London Math. Soc. (2) 62 (2000), no. 3, 784–794. 


