Is there a 0-1 law for the theory of groups?

Question

For each first order sentence $\phi$ in the language of groups, define :

$$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } G}{\text{number of nonisomorphic groups of order} \le N}$$

Thus, $p_N(\phi)$ can be regarded as the probability that $\phi$ is valid in a randomly chosen group of order $\le N$.

Now define $$p(\phi)=\lim_{N \to \infty}p_N(\phi)$$ if this limit exists.

We say that the theory of groups fulfills a first order zero-one law if for every sentence $\phi$, $p(\phi)$ exists and equals either $0$ or $1$. I'm asking myself whether this 0-1 law holds indeed in group theory.

Since it is conjectured that "almost every group is a 2-group", statements like $\exists x: x\ne 1 \wedge x^2=1 \wedge \forall y:xy=yx$ (meaning $2|Z(G)$) or $\forall x: x^3=1 \to x=1$ (no element has order 3) should have probability $1$ and I don't see any possibility to construct any sentence with $p\not \in \{0,1\}$. Am I missing an obvious counterexample, or can you show (under the condition that almost every group is indeed a 2-group) that the theory of finite groups fulfills this 0-1 law?

Answer 1

Sentence proposals:

1. The cardinality of $G$ is even.

   • I strongly suspect the limit diverges by having a zero $\lim\inf$ and a unit $\lim \sup$. Via supermultiplicity, the number of groups of order $2^k n$ is bounded below by the number of order $2^k$ times the number of order $n$, so is infrequently "small" ($n$ prime) and frequently large. Additionally, the number of groups of a given order is upper bounded (see http://www.jstor.org/stable/2946623 ) by something not too stupendously rapidly growing, so no $N$ is going to miraculously overwhelm the running average.

2. Let $P(n)$ be the cardinality of the set of isomorphism classes of groups of order $n$. For all positive integers $n$, $P(n)>0$ since there is at least a cyclic group of order $n$. Let $C(n) = \sum_{i=1}^n P(n)$, which is clearly positive and monotonically increasing on the positive integers. Set $a_0=1$ and, for $j>0$, set $a_j = \min_i\{i \mid C(i)-a_{j-1} > 2^{2j}\}$, such an $i$ exists because $C$ is monotonically increasing. Let $T = \bigcup_{j=0}^\infty [a_{2j}, a_{2j+1})$ and let $\phi$ be the predicate $|G| \in T$. By construction, $p_N(\phi)$ swings by a factor of two through each interval $[a_j, a_{j+1})$ (from something < $2^{-j}$ to something $> 1- 2^{-j}$ and vice versa). Therefore, the limit as $N$ goes to infinity does not exist.

   • It's not clear to me how this sort of see-saw argument fails in any set that has something like a positive semidefinite inverse norm with infinite support. ($P(n)$ is that inverse norm here, taking cardinalities (a usual norm) to the number of isomorphism classes.)