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Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups. A discrete normal subgroup of a connected group G necessarily lies in the center of G and is therefore abelian. Other properties: every discrete group is totally disconnected; every subgroup of a discrete group is discrete.
Pages in category "Discrete groups" The following 26 pages are in this category, out of 26 total. This list may not reflect recent changes. ...
Small groups of prime power order p n are given as follows: Order p: The only group is cyclic. Order p 2: There are just two groups, both abelian. Order p 3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p 2 by a cyclic group of order p.
A periodic wallpaper pattern gives rise to a wallpaper group. Examples and applications of groups abound. A starting point is the group of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups.
Equivalently, a locally profinite group is a topological group that is Hausdorff, locally compact, and totally disconnected. Moreover, a locally profinite group is compact if and only if it is profinite; this explains the terminology. Basic examples of locally profinite groups are discrete groups and the p-adic Lie groups.
Lattices are best thought of as discrete approximations of continuous groups (such as Lie groups). For example, it is intuitively clear that the subgroup of integer vectors "looks like" the real vector space in some sense, while both groups are essentially different: one is finitely generated and countable, while the other is not finitely generated and has the cardinality of the continuum.
This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups of the plane: 2 families of rosette groups – 2D point groups; 7 frieze groups – 2D line ...
Alain Connes used discrete groups with property (T) to find examples of type II 1 factors with countable fundamental group, so in particular not the whole of positive reals +. Sorin Popa subsequently used relative property (T) for discrete groups to produce a type II 1 factor with trivial fundamental group.