The Banach-Tarski paradox
... point (g.x = x and g 6= e) then X is G-paradoxical. In particular if a free group F of rank to acts on X with no non trivial fixed points then X is paradoxical with respect to F ". We are not going to prove this but just give a strategy for the proof. First we construct a set M by using the axiom of ...
... point (g.x = x and g 6= e) then X is G-paradoxical. In particular if a free group F of rank to acts on X with no non trivial fixed points then X is paradoxical with respect to F ". We are not going to prove this but just give a strategy for the proof. First we construct a set M by using the axiom of ...
09 finite fields - Math User Home Pages
... factor of x, all elements of L are roots of xp − x = 0. Thus, with L sitting inside the fixed algebraic closure E of Fp , since a degree pn equation has at most pn roots in E, the elements of L must be just the field K constructed earlier. [5] This proves uniqueness (up to isomorphism). [6] Inside a ...
... factor of x, all elements of L are roots of xp − x = 0. Thus, with L sitting inside the fixed algebraic closure E of Fp , since a degree pn equation has at most pn roots in E, the elements of L must be just the field K constructed earlier. [5] This proves uniqueness (up to isomorphism). [6] Inside a ...
Classify each triangle by its side lengths and angle measurements
... include the headboard of a bed, dressers, chairs, couches, and place mats. Ask him to show where the line of symmetry would be and what makes it a line of symmetry. Be careful of objects such as doors and windows. They could have a line of symmetry, but if there’s a knob or crank on just one side, t ...
... include the headboard of a bed, dressers, chairs, couches, and place mats. Ask him to show where the line of symmetry would be and what makes it a line of symmetry. Be careful of objects such as doors and windows. They could have a line of symmetry, but if there’s a knob or crank on just one side, t ...
An Irrational Construction of R from Z
... The ProofPower-HOL theory was developed in November 2000 by the present author using some novel methods intended to make use of the fact that ProofPowerHOL already possessed reasonably well-supported theories for both natural number and integer arithmetic. As is the norm in the HOL community, the t ...
... The ProofPower-HOL theory was developed in November 2000 by the present author using some novel methods intended to make use of the fact that ProofPowerHOL already possessed reasonably well-supported theories for both natural number and integer arithmetic. As is the norm in the HOL community, the t ...
Some Cardinality Questions
... a unique F -vector space of dimension α, say i∈α F . Recall that the infinite direct sum is the subgroup of the direct product consisting of tuples all but finitely many of which are zero, so that the cardinality of an α-dimensional F -vector space is equal to |F |α if F and α are finite, and otherw ...
... a unique F -vector space of dimension α, say i∈α F . Recall that the infinite direct sum is the subgroup of the direct product consisting of tuples all but finitely many of which are zero, so that the cardinality of an α-dimensional F -vector space is equal to |F |α if F and α are finite, and otherw ...
Document
... seemingly keen for a lesson. I said, "Tap eight." She did a brilliant exhibition, first tapping it in 4, 4, then giving me a hasty glance and doing it in 2, 2, 2, 2, before coming for her nut. It is astonishing that Star learned to count up to 8 with no difficulty, and of her own accord discovered t ...
... seemingly keen for a lesson. I said, "Tap eight." She did a brilliant exhibition, first tapping it in 4, 4, then giving me a hasty glance and doing it in 2, 2, 2, 2, before coming for her nut. It is astonishing that Star learned to count up to 8 with no difficulty, and of her own accord discovered t ...
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.