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Page "Word problem for groups" ¶ 32
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Some Related Sentences

Every and finitely
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 finitely generated ideal of A is principal ( i. e., A is a B├ęzout domain ) and A satisfies the ascending chain condition on principal ideals.
Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense.
Every finitely generated ideal of a Boolean ring is principal ( indeed, ( x, y )=( x + y + xy )).
* Every left ideal I in R is finitely generated, i. e. there exist elements a < sub > 1 </ sub >, ..., a < sub > n </ sub > in I such that I = Ra < sub > 1 </ sub > + ... + Ra < sub > n </ sub >.
Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented.
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 >.
" I can now state the physical version of the Church-Turing principle: ' Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means.
* Every substructure is the union of its finitely generated substructures ; hence Sub ( A ) is an algebraic lattice.
A finitely-generated abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form for some prime number p and some positive integer n. Every finitely-generated abelian group is a direct sum of ( finitely many ) indecomposable abelian groups.
This is up to isomorphism the only indecomposable module over R. Every left R-module is a direct sum of ( finitely or infinitely many ) copies of this module K < sup > n </ sup >.
Every ( bounded ) convex polytope is the image of a simplex, as every point is a convex combination of the ( finitely many ) vertices.

Every and generated
Every simple module is cyclic, that is it is generated by one element.
Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by
Every lattice in can be generated from a basis for the vector space by forming all linear combinations with integer coefficients.
Every film today, whether it be live-action, computer generated, or traditional hand-drawn animation is made up of hundreds of individual shots that are all placed together during editing to form the single film that is viewed by the audience.
A large part of the interest generated was due to the eclectic mix of samples and influences, evident on Every Man and Woman, which was dedicated to Dewey Bunnell of the band America.
* Every compact space is compactly generated.
* Every locally compact space is compactly generated.
* Every first-countable space is compactly generated.
* Every CW complex is compactly generated Hausdorff.
Every ticket generated by the system has persistence or " history " showing what happened to the ticket within its life cycle.
Every profile also captures some non-crime statistics: the amount of overtime generated by members of the command, the number of department vehicle accidents, absence rates due to sick time and line-of-duty injuries, and the number of civilian complaints lodged against members of the unit.

Every and group
Every finite simple group is isomorphic to one of the following groups:
Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.
* Every group G acts on G, i. e. in two natural but essentially different ways:, or.
Every galaxy of sufficient mass in the Local Group has an associated group of globular clusters, and almost every large galaxy surveyed has been found to possess a system of globular clusters.
* Every Lie group is parallelizable, and hence an orientable manifold ( there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity )
Every instrumental group ( or section ) has a principal who is generally responsible for leading the group and playing orchestral solos.
* Every closed subgroup of a profinite group is itself profinite ; the topology arising from the profiniteness agrees with the subspace topology.
While he admits the existence of caste-based discrimination, he writes that " Every social group cannot be regarded as a race simply because we want to protect it against prejudice and discrimination ".
Every local group is required to have a seneschal who reports to the kingdom's seneschal.
Every local group is required to have one.
* Every topological group is completely regular.
Every summer the group gathers in Newport, RI for week long dance training, seaside teas, and evenings enjoying the splendors of the Gilded Age.
Every synset contains a group of synonymous words or collocations ( a collocation is a sequence of words that go together to form a specific meaning, such as " car pool "); different senses of a word are in different synsets.
Every group can be trivially made into a topological group by considering it with the discrete topology ; such groups are called discrete groups.
Every topological group can be viewed as a uniform space in two ways ; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.
Every subgroup of a topological group is itself a topological group when given the subspace topology.
:" Every group is naturally isomorphic to its opposite group "
Every ten years, when the general census of population takes place, each citizen has to declare which linguistic group they belong or want to be aggregated to.

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