Write sentences ϕ in the language of groups

Question

Don't know where and how to start.

Write sentences ϕ in the language of groups, i.e, the language with

• binary function symbol ·, interpreted as multiplication,

• unary function symbol −1, interpreted as inversion,

• and constant symbol e, which is interpreted as the identity element,

1. G entails ϕ if and only if G is isomorphic to the symmetric group on three letters.

2. G entails ϕ if and only if G is abelian; if and only if G is nonabelian. Draw the parsing trees for each formula, underline all subformulas and indicate their free variables.

3. G entails ϕ if and only if G has an element of order n; if and only if every element of G has order n.

4. G entails ϕ if and only if G has a nontrivial central element; if and only if G has trivial center.

Answer 1

Hint

For each bullett, we have to write a sentence $\phi$ such that :

$G \vDash \phi \text { iff } G \text { is a Group with the specified propertry }$.

We can start writing a single sentence $\text {Grax}$ specifying the Group axioms :

$\text {Grax} := [∀a, b, c (a \circ (b \circ c) = (a \circ b) \circ c)] \land [ ∃e∀a (e \circ a = a) \land (a \circ e = a)] \land [ ∃e ∀a ∃b (a \circ b = e) \land (b \circ a=e)]$.

Having done this, we can consider e.g. the abelian case, writing the formula specifying Commutativity :

$\text {Comm} := ∀a∀b(a \circ b = b \circ a)$.

Thus, a group $G$ is abelian iff :

$G \vDash \text {Grax} \land \text {Comm}$.