 Page "Classification of finite simple groups" ¶ 2
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Every and finite ** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion. # Every open cover of A has a finite subcover. * Countably compact: Every countable open cover has a finite subcover. # Every finite and contingent being has a cause. Hilbert's example: " the assertion that either there are only finitely many prime numbers or there are infinitely many " ( quoted in Davis 2000: 97 ); and Brouwer's: " Every mathematical species is either finite or infinite. * Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way. Every finite tree structure has a member that has no superior. Every rational number / has two closely related expressions as a finite continued fraction, whose coefficients can be determined by applying the Euclidean algorithm to. * Every finite tree with n vertices, with, has at least two terminal vertices ( leaves ). Every finite group of exponent n with m generators is a homomorphic image of B < sub > 0 </ sub >( m, n ). Every known Sierpinski number k has a small covering set, a finite set of primes with at least one dividing k · 2 < sup > n </ sup >+ 1 for each n > 0. Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space. : Every oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume. Every finite or bounded interval of the real numbers that contains an infinite number of points must have at least one point of accumulation. Every field of either type can be realized as the field of fractions of a Dedekind domain in which every non-zero ideal is of finite index. Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero. Every finite group has a composition series, but not every infinite group has one. * Every finite or cofinite subset of the natural numbers is computable. * Every subset of may be covered by a finite set of positive orthants, whose apexes all belong to * Every finite subextension of F / k is separable. Every finite ordinal ( natural number ) is initial, but most infinite ordinals are not initial. * Every finite-dimensional central simple algebra over a finite field must be a matrix ring over that field. * Every commutative semisimple ring must be a finite direct product of fields.

Every and simple # Every simple path from a given node to any of its descendant leaves contains the same number of black nodes. Every simple R-module is isomorphic to a quotient R / m where m is a maximal right ideal of R. By the above paragraph, any quotient R / m is a simple module. Every simple module is indecomposable, but the converse is in general not true. Every simple module is cyclic, that is it is generated by one element. " I might easily have written this story in the traditional manner [...] Every novelist knows the recipe [...] It is not very difficult to follow a simple, chronological scheme which the critics will understand [...] But I, after all, am trying to tell the story of this Chapelizod family in a new way. Every October, Moriarty plays host to the Pinto Bean Fiesta, which is composed of a bunch of simple games in Crossly Park, as well as a parade and crowning of a " Pinto Bean Queen. Every one of the infinitely many vertices of G can be reached from v < sub > 1 </ sub > with a simple path, and each such path must start with one of the finitely many vertices adjacent to v < sub > 1 </ sub >. Every hour that Napoleon could have attacked earlier as he did, would have been is his favour, but the French could not attack in the morning for the simple reason that the entire army had not yet taken its battle positions. Every maximal outerplanar graph is the visibility graph of a simple polygon. Every closed curve c on X is homologous to for some simple closed curves c < sub > i </ sub >, that is, * Every finite-dimensional simple algebra over R must be a matrix ring over R, C, or H. Every central simple algebra over R must be a matrix ring over R or H. These results follow from the Frobenius theorem. * Every finite-dimensional simple algebra over C must be a matrix ring over C and hence every central simple algebra over C must be a matrix ring over C. * Every automorphism of a central simple algebra is an inner automorphism ( follows from Skolem – Noether theorem ). * Every 4-dimensional central simple algebra over a field F is isomorphic to a quaternion algebra ; in fact, it is either a two-by-two matrix algebra, or a division algebra. Every simple ring R with unity is both left and right primitive. # Personal right: Every person has a right to life but this right is restricted and has attached certain duties – simple living is essential. Every Sámi settlement had its seita, which had no regular shape, and might consist of smooth or odd-looking stones picked out of a stream, of a small pile of stones, of a tree-stump, or of a simple post. * Rule 2 Each column has a simple generic model: Every column can have its own meta-model

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