List of First-order Theories - Groups

Groups

The signature of group theory has one constant 1 (the identity), one function of arity 1 (the inverse) whose value on t is denoted by t−1, and one function of arity 2, which is usually omitted from terms. For any integer n. tn is an abbreviation for the obvious term for the nth power of t.

Groups are defined by the axioms

• Identity: ∀x 1x = x ∧ x1 = x
• Inverse: ∀x x−1x = 1 ∧ xx−1 = 1
• Associative: ∀x∀y∀z (xy)z = x(yz)

Some properties of groups that can be defined in the first-order language of groups are:

• Abelian ∀x ∀y xy = yx.
• Torsion free ∀x x2 = 1→x = 1, ∀x x3 = 1 → x = 1, ∀x x4 = 1 → x = 1, ...
• Divisible ∀x ∃y y2 = x, ∀x ∃y y3 = x, ∀x ∃y y4 = x, ...
• Infinite (as in identity theory)
• Exponent n (for any fixed positive integer n) ∀x xn = 1
• Nilpotent of class n (for any fixed positive integer n)
• Solvable of class n (for any fixed positive integer n)

The theory of Abelian groups is decidable. The theory of Infinite divisible torsion-free abelian groups is complete, as is the theory of Infinite abelian groups of exponent p (for p prime).

The theory of finite groups is the set of first-order statements in the language of groups that are true in all finite groups (there are plenty of infinite models of this theory). It is not completely trivial to find any such statement that is not true for all groups: one example is "given two elements of order 2, either they are conjugate or there is a non-trivial element commuting with both of them".

The properties of being finite, or free, or simple, or torsion are not first-order. More precisely, the first-order theory of all groups with one of these properties has models that do not have this property.