What is connection between Group theory and Logic? I have just started to study Algebraic logic.

What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems) wiki

For Classical sentential logic the relevant algebraic structure is Boolean algebra.
Is there logical system with a Group (algebraic structure) as a model (interpretations)?
 A: This is an answer to the more general question in the title of your post.
There are many undecidability results in group theory, many of which relate to groups defined by finite presentation. In general you cannot decide  whether such a group is trivial, finite, abelian, etc. Also there are specific finite presentations for which the group defined has unsolvable word problem, meaning that you cannot decide whether two words in the group generators represent the same group element. The proofs of all of these results involve results from logic (but typically also results from group theory - HNN extensions are involved in the proof of the unsolvability of the word problem).
What is particularly interesting are the results that connect the algebraic properties of a group with their decidability properties. For example, the Boone-Higman Theorem says that a finitely presented group has solvable word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group.
