Glossary of Group Theory - Finiteness Conditions

Finiteness Conditions

The order |G| (or o(G)) of a group is the cardinality of G. If the order |G| is (in-)finite, then G itself is called (in-)finite. An important class is the group of permutations or symmetric groups of N letters, denoted S_N. Cayley's theorem exhibits any finite group G as a subgroup of the symmetric group on G. The theory of finite groups is very rich. Lagrange's theorem states that the order of any subgroup H of a finite group G divides the order of G. A partial converse is given by the Sylow theorems: if pn is the greatest power of a prime p dividing the order of a finite group G, then there exists a subgroup of order pn, and the number of these subgroups is also known. A projective limit of finite groups is called profinite. An important profinite group, fundamental for p-adic analysis, class field theory, and l-adic cohomology is the ring of p-adic integers and the profinite completion of Z, respectively

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Most of the facts from finite groups can be generalized directly to the profinite case.

Certain conditions on chains of subgroups, parallel to the notion of Noetherian and Artinian rings, allow to deduce further properties. For example the Krull-Schmidt theorem states that a group satisfying certain finiteness conditions for chains of its subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.

Another, yet slightly weaker, level of finiteness is the following: a subset A of G is said to generate the group if any element h can be written as the product of elements of A. A group is said to be finitely generated if it is possible to find a finite subset A generating the group. Finitely generated groups are in many respects as well-treatable as finite groups.