Finiteness properties of groups information

study the group. It is mostly of interest for the study of infinite groups. Special cases of groups with finiteness properties are finitely generated...

finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of...

group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime...

of reflection groups, and finite Coxeter groups were classified in 1935. Coxeter groups find applications in many areas of mathematics. Examples of finite...

takes its place. Most properties of representations of finite groups can be transferred with appropriate changes to compact groups. For this we need a counterpart...

MR 0004123. Bestvina, Mladen; Brady, Noel (1997). "Morse theory and finiteness properties of groups". Inventiones Mathematicae. 129 (3): 445–470. Bibcode:1997InMat...

"synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example...

properties of finitely generated groups and topological and geometric properties of spaces on which these groups act. The word problem for a finitely...

of group with a given such property: finite groups, periodic groups, simple groups, solvable groups, and so on. Rather than exploring properties of an...

version of discrete Morse theory for cubical complexes and applied it to study homological finiteness properties of subgroups of right-angled Artin groups. In...

transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups. The Fourier transform of a function...

specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive...

trivial. A group is residually finite if and only if it can be embedded inside the direct product of a family of finite groups. Examples of groups that are...

periodic groups and finite groups, when only finitely generated groups are considered: Does specifying an exponent force finiteness? The existence of infinite...

many finitely generated recursively presented groups. Bernhard Neumann has shown that there are uncountably many non-isomorphic two generator groups. Therefore...

retrieved 31 May 2011 Brown, Kenneth S. (February 1987), "Finiteness properties of groups" (PDF), Journal of Pure and Applied Algebra, 44 (1–3): 45–75, doi:10...

and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected...

finite group whose p-Sylow subgroups are cyclic is a semidirect product of two cyclic groups, in particular solvable. Such groups are called Z-groups...

finite reflection group. In fact it turns out that most finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups,...

classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic...

universal property. Free groups first arose in the study of hyperbolic geometry, as examples of Fuchsian groups (discrete groups acting by isometries on...

subgroup of a Hausdorff group is closed. every discrete subgroup of a compact Hausdorff group is finite. Frieze groups and wallpaper groups are discrete...

permutation groups is Burnside's Theory of Groups of Finite Order of 1911. The first half of the twentieth century was a fallow period in the study of group theory...

Symmetric group. As finite symmetric groups are the groups of all permutations of a set with finite elements, and the alternating groups are groups of even...

still have many of the finiteness properties of polycyclic groups; for example, they satisfy the maximal condition, and they are finitely presented and...

also within the group). Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in...