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Fuchsian groups are, by definition, discrete subgroups of the isometry group of the hyperbolic plane. A Fuchsian group that preserves orientation and acts on the upper half-plane model of the hyperbolic plane is a discrete subgroup of the Lie group PSL(2, R ), the group of orientation preserving isometries of the upper half-plane model of the ...
Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R. As a simple example, the cyclic group of order n has the ...
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.
In biology, a cline is a measurable gradient in a single characteristic (or biological trait) of a species across its geographical range. [1] Clines usually have a genetic (e.g. allele frequency, blood type), or phenotypic (e.g. body size, skin pigmentation) character.
The science of classification, in biology the arrangement of organisms into a classification [4] "The science of classification as applied to living organisms, including the study of means of formation of species, etc." [5] "The analysis of an organism's characteristics for the purpose of classification" [6]
The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center. Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows. Write
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Examples of groups that do not have property (T) include The additive groups of integers Z, of real numbers R and of p-adic numbers Q p. The special linear groups SL(2, Z) and SL(2, R), as a result of the existence of complementary series representations near the trivial representation, although SL(2,Z) has property (τ) with respect to ...