soft question - How to distinguish property of particular representation from property of algebraic structure?

I'm not sure I completely understand what you're asking, but here is some information that appears to be relevant.

In the context you're describing, you have two languages: the pure language L₀ of groups and the augmented language L₁ of groups together with a linear representation over some field (see note). You seem to be asking the following:

• When does a sentence in L₁ (i.e. a property of groups with linear representations) equivalent to a sentence in L₀ (i.e. a pure property of groups)?

The Robinson Consistency Theorem answers this, at least in part.

Let G be a group and let L₂ be the language of groups augmented with a constant for every element of G. Let T₂ be the complete theory of G in L₂ and let T₀ = T₂ ∩ L₀ be the complete theory of G in L₀. The difference between T₀ and T₂ is that statements in T₂ are allowed to mention particular elements of G. So if x is an element of order 2 in G, then that fact is recorded in T₂ but not in T₀. However, the fact that G has an element of order 2 is recorded in T₀ since that fact does not explicitly mention any element of G.

Now let φ be any statement in the language L₁ of groups with linear representations. The Robinson Consistency Theorem says that if T₀ ∪ T₁ ∪ {φ} is consistent then so is T₂ ∪ T₁ ∪ {φ}, where T₁ ⊆ L₁ is the theory of linear representations (see note). Stated differently, T₂ ∪ T₁ ⊦ φ if and only if T₀ ∪ T₁ ⊦ φ. (Recall that T ⊦ φ iff T ∪ {¬φ} is inconsistent.) The models of T₂ are precisely the elementary extensions of G. Thus the following are equivalent:

• φ is a consequence of some purely group theoretic property of G


• φ is true for every linear representation of an elementary extension of G

Sometimes we can say more. If φ is an existential statement in L₁ (i.e. φ is equivalent to a sentence of the form ∃x,y,z,...φ₀(x,y,z,...) where φ₀ is quantifier free) then we don't have to check all elementary extensions of G. Thus, for such φ, the following are equivalent:

• φ is true in all linear representations of G


• φ is a consequence of some purely group theoretic property of G

Of course, the Robinson Consistency Theorem is not particular to groups and linear representations of groups; the same reasoning applies in all sorts of contexts.

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Note: I'm assuming that all languages are first-order (possibly with multiple sorts). There are various ways to formulate linear representations in first-order logic, but none are completely satisfactory. For fixed dimension n, one can add a sort F for the field elements together with functions a_i,j:G→F for the entries of the matrices of the representations. Then T₁ consists of all field axioms together with all required identities between the a_i,j.