The set of all continuous deformation congruence classes of paths in a manifold is one type of these objects.
Equivalent descriptions of these objects can be passed between by a finite number of Tietze transformations. The set of all continuous deformation congruence classes of paths in a manifold is one type of these objects. The largest prime power that divides the order of one of these objects is also the order of one of its subsets according to one of Sylow's theorems, which is a partial converse of Lagrange's theorem. Every finite one is isomorphic to a subset of the symmetric one according to Cayley's theorem. Sometimes defined as a monoid with an inverse element, these objects are a set with an identity element and an inverse for every element that is closed under a binary operation which is associative. For 10 points, name these objects that, when their operation possesses commutativity, are known as "abelian," and which are studied in abstract algebra along with rings and fields.
Equivalent descriptions of these objects can be passed between by a finite number of Tietze transformations.
groups