Show this language structure models this sentence.

Question

In an effort to educate myself, I am attempting the second problem in first chapter of the book "Model Theory" by Marker. The problem is reproduced below:

Let $\mathcal{L} = \{\cdot, e\}$ be the language of groups. Show that there is a sentence $\phi$ such that $\mathcal{M} \models \phi$ if and only if $\mathcal{M} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/ 2 \mathbb{Z}$.

I have two problems.

1. Syntactically speaking, what does the symbol $\cong$ express?

2. Could someone walk me through a formal proof of this claim so I know how "it's done"?

Answer 1

You want a sentence $\phi$ to describe exactly that your group is $\mathbb Z/{2\mathbb Z}\times Z/{2\mathbb Z}$.

In english, you could say this by:

• there exists exactly four elements $a,b,c,d$
• the group axioms are verified
• for all element $x$, we have $x\cdot x=e$.

Since the only groups of order $4$ are $\mathbb Z/{2\mathbb Z}\times Z/{2\mathbb Z}$ and $\mathbb Z/{4\mathbb Z}$, you are done.

Try to express the $3$ conditions using logical formalism. For instance the last one is $\forall x, x\cdot x=e$.

Answer 2

It is certainly possible to try to find some interesting property of that group that characterizes it. But you can also approach the problem in a very straightforward way. You can simply write down that there exist three elements $a$, $b$ and $c$ such that:

1. $a$, $b$ and $c$ are distinct, and distinct from $e$;

2. Every element of the structure is equal to one of $a$, $b$, $c$, $e$; and

3. The elements $a$, $b$, $c$ and $e$ satisfy the appropriate multiplication table.

Any finite structure in a finite language can be characterized up to isomorphism in a single sentence by this method.