[edit] Star polyhedra

... Regular if it is vertex-transitive, edge-transitive and face-transitive (this implies that every face is the same regular polygon; it also implies that every vertex is regular). Quasi-regular if it is vertex-transitive and edge-transitive (and hence has regular faces) but not face-transitive. A quas ...

... Regular if it is vertex-transitive, edge-transitive and face-transitive (this implies that every face is the same regular polygon; it also implies that every vertex is regular). Quasi-regular if it is vertex-transitive and edge-transitive (and hence has regular faces) but not face-transitive. A quas ...

Neutrosophic Triplet Group

... Groups are so much important in algebraic structures and they play the role of back bone in almost all algebraic structures. Groups are count in older algebraic structures. It is so much rich algebraic structure than any other structure. In many algebraic structures, groups are present concrete foun ...

... Groups are so much important in algebraic structures and they play the role of back bone in almost all algebraic structures. Groups are count in older algebraic structures. It is so much rich algebraic structure than any other structure. In many algebraic structures, groups are present concrete foun ...

Maximal compact subgroups in the o-minimal setting

... (i) G has maximal definable definably compact subgroups. (ii) G = K · N (G), for some definable definably compact subgroup K < G. (iii) A(G) = {e}. Proof. (i) ⇒ (iii): Take K to be a maximal definable definably compact subgroup, and T a 0-Sylow of G. Suppose, for a contradiction, that A(G) is infini ...

... (i) G has maximal definable definably compact subgroups. (ii) G = K · N (G), for some definable definably compact subgroup K < G. (iii) A(G) = {e}. Proof. (i) ⇒ (iii): Take K to be a maximal definable definably compact subgroup, and T a 0-Sylow of G. Suppose, for a contradiction, that A(G) is infini ...

Intro Abstract Algebra

... sort of rule which to an input s 2 S \computes" or \associates" an output f (s) 2 T , this way of talking is inadequate, for many reasons. Rather, the formal (possibly non-intuitive) de nition of function f from a set S to a set T is that it is a subset G of the cartesian product S T with the prop ...

... sort of rule which to an input s 2 S \computes" or \associates" an output f (s) 2 T , this way of talking is inadequate, for many reasons. Rather, the formal (possibly non-intuitive) de nition of function f from a set S to a set T is that it is a subset G of the cartesian product S T with the prop ...

National Curriculum Glossary. - Bentley Heath Church Of England

... This glossary has been developed by the National Centre for Excellence in the Teaching of Mathematics (NCETM) in response to a request from the Department for Education to support the publication of the new national curriculum for mathematics which will be implemented in schools in September 2014. T ...

... This glossary has been developed by the National Centre for Excellence in the Teaching of Mathematics (NCETM) in response to a request from the Department for Education to support the publication of the new national curriculum for mathematics which will be implemented in schools in September 2014. T ...

H1 Angles and Symmetry

... Then find a. The angle a and the angle marked 70° are opposite angles, so a = 70° . The angles a and b are alternate angles so a = b = 70° . The angles b and c are opposite angles so b = c = 70° . The angles a and d are a pair of interior angles, so a + d = 180° , but a = 70° , so 70° + d = 180° d = ...

... Then find a. The angle a and the angle marked 70° are opposite angles, so a = 70° . The angles a and b are alternate angles so a = b = 70° . The angles b and c are opposite angles so b = c = 70° . The angles a and d are a pair of interior angles, so a + d = 180° , but a = 70° , so 70° + d = 180° d = ...

Here - Personal.psu.edu

... 1.1.4 Show that if a|bc then it is not necessarily true that a|b or a|c. For example 8|(4)(2) but 8 6 |4 and 8 6 |2 1.1.5 Show that if a|c and b|c then it is not necessarily true that ab|c. For example, 2|4 and 4|4, but 2(4) = 8 6 |4. 1.1.6 If (a, c) = 1 = (b, c), show that (ab, c) = 1. Assume an in ...

... 1.1.4 Show that if a|bc then it is not necessarily true that a|b or a|c. For example 8|(4)(2) but 8 6 |4 and 8 6 |2 1.1.5 Show that if a|c and b|c then it is not necessarily true that ab|c. For example, 2|4 and 4|4, but 2(4) = 8 6 |4. 1.1.6 If (a, c) = 1 = (b, c), show that (ab, c) = 1. Assume an in ...

# Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation; the addition of any two integers forms another integer. The abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Point groups are used to help understand symmetry phenomena in molecular chemistry; and Poincaré groups can express the physical symmetry underlying special relativity.The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A theory has been developed for finite groups, which culminated with the classification of finite simple groups announced in 1983. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.