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Every subgroup of a free abelian group is itself free abelian, which is important for the description of a general abelian group as a cokernel of a homomorphism between free abelian groups.
Some Related Sentences
Every and subgroup
* Every closed subgroup of a profinite group is itself profinite ; the topology arising from the profiniteness agrees with the subspace topology.
* 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 subgroup of a topological group is itself a topological group when given the subspace topology.
Every open subgroup H is also closed, since the complement of H is the open set given by the union of open sets gH for g in G
Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by
Iwasawa worked with so-called-extensions: infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form, so by Galois theory, a-extension is the same thing as a tower of fields such that.
* Every triangle group T is a discrete subgroup of the isometry group of the sphere ( when T is finite ), the Euclidean plane ( when T has a Z + Z subgroup of finite index ), or the hyperbolic plane.
Recall that a subsemigroup G of a semigroup S is a subgroup of S ( also called sometimes a group in S ) if there exists an idempotent e such that G is a group with identity element e. A semigroup S is group-bound if some power of each element of S lies in some subgroup of S. Every finite semigroup is group-bound, but a group-bound semigroup might be infinite.
Every subgroup is organized around a set of brothers, each of whom is often married to a group of sisters.
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.
Every proper subgroup of G can be assumed a solvable group, meaning that much theory of such subgroups could be applied.
Every quasinormal subgroup is a modular subgroup, that is, a modular element in the lattice of subgroups.
Every group has itself ( the improper subgroup ) and the trivial subgroup as two of its fully characteristic subgroups.
Every and free
Every free man, born in lawful wedlock, and neither excommunicate nor outlaw, was eligible for membership.
Every other province had the freedom to regulate the religious question as it wished, although the Union stated every person should be free in the choice of his personal religion and no person should be prosecuted based on his or her religious choice.
The Body of Liberties adopted in 1641 by the Massachusetts Bay colonists states, “ Every married woman shall be free from bodily correction or stripes by her husband, unless it be in his own defense from her assault .” In the United States, legal decisions in Mississippi ( 1824 ) and North Carolina ( 1868 and 1874 ) make reference to — and reject — an unnamed " old doctrine " or " ancient law " by which a man was allowed to beat his wife with a stick no wider than his thumb.
Every antiseptic, however good, is more or less toxic and irritating to a wounded surface ; as a result, in surgery, the antiseptic method has been replaced by aseptic method, which is preventative in nature and relies on keeping free from the invasion of bacteria rather than destroying them when present.
* The Empty Space: Every Friday the Students ' Union hosts a live music venue in The Empty Space, attended by students for free ( this was canceled in 2011 due to changes installed by the 69th Student Legislative Council ).
* Every August the residents of Sleepy Eye host the annual " Corn Days " event, where free buttered corn is provided, as well as live music, a flea market, parade, and various other events.
* Every free abelian group is torsion-free, but the converse is not true, as is shown by the additive group of the rational numbers Q.
Every week of the Michaelmas and Lent terms, and twice in Easter term, Churchill is host to Pav, a music event unusual for Cambridge events in that it is free and open to all university members.
Every summer, a free tourist train in the city centre connects the ancient parts of the city with the government district.
Every town wanted to be a railway centre, and the government had great confidence in the ability of the free market to provide low freight rates to the province's farmers provided sufficient charters were issued to competing companies.
Every month Hustler is mailed, uninvited and for free, to some of the offices of Members of the United States Congress.
Every Riemann surface is the quotient of a free, proper and holomorphic action of a discrete group on its universal covering and this universal covering is holomorphically isomorphic ( one also says: " conformally equivalent ") to one of the following:
Every vector space is free, and the free vector space on a set is a special case of a free module on a set.
Every Ontario resident with his or her primary and permanent home in Ontario is entitled to access emergency and preventive medical care ( although Bariatric surgery in many cases is not covered ) under OHIP free of charge.
Every and abelian
Every algebraic curve C of genus g ≥ 1 is associated with an abelian variety J of dimension g, by means of an analytic map of C into J.
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.
Every abelian category is an exact category if we just use the standard interpretation of " exact ".