 Solution

We describe a collection of finite multiplication patterns. For finite setswriteCall $(E,D)$ $T$-forbidden if the following theory, in $L$ expanded by unary symbols $f_1,\dots ,f_n$, is inconsistent:

Let $T^{\ast }$ contain the group axioms and, for every $T$-forbidden finite pattern, the sentenceThis is a first-order theory in the language of groups.

Suppose $G$ is $T$-good. Choose $M\models T$ and an embedding $\rho :G\hookrightarrow \mathrm{Aut}(M)$. If a tuple $(g_1,\dots ,g_n)$ in $G$ realized a forbidden pattern, interpreting $f_i$ as $\rho (g_i)$ would give a model of its supposedly inconsistent automorphism theory. Thus no such tuple exists, and $G\models T^{\ast }$.

Conversely, suppose $G$ is $T$-bad. By part (e), there is a finite set $X=\{g_1,\dots ,g_n\}\subseteq G$ for which $T_G\cap L_X$ is inconsistent. LetThe associated automorphism theory is precisely $T_G\cap L_X$, up to renaming its function symbols, so $(E,D)$ is $T$-forbidden. But $(g_1,\dots ,g_n)$ realizes $P_{E,D}$ in $G$, and therefore $G$ violates the corresponding forbidding axiom. Hence $G/\models T^{\ast }$.

We have proved