group is said to be locally P if every finitely generated subgroup is P. For instance, a group is locally finite if every finitely generated subgroup is finite, and a group is locally soluble if every finitely generated subgroup is soluble. Properties of finite groups For finite groups, a "small neighborhood" is taken to be a subgroup defined in terms of a prime number p, usually the local subgroups, the normalizers of the nontrivial p-subgroups. In which case, a property is said to be local if it can be detected from the local subgroups. Global and local properties formed a significant portion of the early work on the classification of finite simple groups, which was carried out during the 1960s. Properties of commutative rings For commutative rings, ideas of algebraic geometry make it natural to take a "small neighborhood" of a ring to be the localization at a prime ideal. In which case, a property is said to be local if it can be detected from the local rings. For instance, being a flat module over a commutative ring is a local property, but being a free module is not. For more, see Localization of a module. See also Local path connectedness References General topology Homeomorphisms