Solution

We describe a collection of finite multiplication patterns. For finite sets

E \subseteq {1, \dots, n}^{3}, D \subseteq {1, \dots, n}^{2},

write

P_{E, D} (x_{1}, \dots, x_{n}) = ⋀_{(i, j, k) \in E} x_{i} x_{j} = x_{k} \land ⋀_{(i, j) \in D} x_{i} \neq = x_{j} .

Call

(E, D)

T

-forbidden if the following theory, in

L

expanded by unary symbols

f_{1}, \dots, f_{n}

, is inconsistent:

Let

T^{*}

contain the group axioms and, for every

T

-forbidden finite pattern, the sentence

\forall x_{1} \dots \forall x_{n} \neg P_{E, D} (x_{1}, \dots, x_{n}) .

This is a first-order theory in the language of groups.

Suppose

G

T

-good. Choose

M ⊨ T

and an embedding

ρ : G ↪ Aut (M)

. If a tuple

(g_{1}, \dots, g_{n})

G

realized a forbidden pattern, interpreting

f_{i}

ρ (g_{i})

would give a model of its supposedly inconsistent automorphism theory. Thus no such tuple exists, and

G ⊨ T^{*}

Conversely, suppose

G

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. Let

E = {(i, j, k) : g_{i} g_{j} = g_{k}}, D = {(i, j) : g_{i} \neq = g_{j}} .

The associated automorphism theory is precisely

T_{G} \cap L_{X}

, up to renaming its function symbols, so

(E, D)

T

-forbidden. But

(g_{1}, \dots, g_{n})

realizes

P_{E, D}

G

, and therefore

G

violates the corresponding forbidding axiom. Hence

G \neq ⊨ T^{*}

We have proved

G ⊨ T^{*} ⟺ G is T -good .

Solved by gpt-5.6-sol high.

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