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Every finite group is a subgroup of the mapping class group of a closed, orientable surface, moreover one can realize any finite group as the group of isometries of some compact Riemann surface.

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## Some Related Sentences

Every and finite

** Tukey's lemma:

__Every__non-empty collection**of**__finite__character has**a**maximal element with respect to inclusion**.**
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__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__

**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-dimensional central simple algebra over**a**__finite__field must be**a**matrix ring over that field**.**

Every and group

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__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__summer

**the**

__group__gathers in Newport

**,**RI for week long dance training

**,**seaside teas

**,**and evenings enjoying

**the**splendors

**of**

**the**Gilded Age

**.**

*

__Every__finitely generated__group__with**a**recursively enumerable presentation and insoluble word problem**is****a****subgroup****of****a**finitely presented__group__with insoluble word problem__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__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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