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Abelian groupsAbelian groups
An abelian group is a structure G = (G, + ,−,0), where + is an infix binary operation, called the
group addition, − is a prefix unary operation, called the
group negative and 0 is a constant (nullary operation), called the additive identity element, such that
+ is commutative: x + y = y + x,
+ is associative: (x + y) + z = x + (y + z),
0 is an additive identity for + : 0 + x = x, and
− gives an additive inverse for + : −x + x = 0.
Let G and H be abelian groups. A morphism from G to H is a function h : G→H that is a homomorphism: h(x + y) = h(x) + h(y).
Remark: It follows that h(−x) = −h(x) and h(0) = 0.
(Z, + , −, 0), the integers, with addition, unary subtraction, and zero. The variety of abelian groups is generated by this algebra.
| Classtype | variety |
| Equational theory | decidable in polynomial time |
| Quasiequational theory | decidable |
| First-order theory | decidable [W. Szmielew, Decision problem in group theory, Library of the Tenth International Congress of Philosophy, Amsterdam, August 11--18, 1948, Vol.1, Proceedings of the Congress (1949) 763--766 MRreview] |
| Locally finite | no |
| Residual size | ω |
| Congruence distributive | no (Z2×Z2) |
| Congruence modular | yes |
| Congruence n-permutable | yes, n = 2, p(x,y,z) = x−y + z |
| Congruence regular | yes, congruences are determined by subalgebras |
| Congruence uniform | yes |
| Congruence types | permutational |
| Congruence extension property | yes, if K ≤ H ≤ G then K ≤ G |
| Definable principal congruences | no |
| Equationally definable principal congruences | no |
| Amalgamation property | yes |
| Strong amalgamation property | yes |
| Epimorphisms are surjective | yes |