 About 305,000 results  books.google.com In fact, given the very large number of groups involved, it was necessary to store
them in a highly compressed format. The insolvable groups are stored as
permutation groups. We shall outline an algorithm to determine up to
isomorphism the ... 

 books.google.com If p and q are primes with p < q, then there exists a nonabelian group of order pq
if and only if p divides q − 1, and in this case the nonabelian group is unique up
to isomorphism. It may be taken to be a semidirect product of the cyclic groups C
... 

 books.google.com Historical note 225 Each of the 17 types is unique up to isomorphism 40 In the
case of groups of types pi, p2, pm, pg, cm, pmm, pmg, pgg and cmm, show that
for any two groups of the same type that generators may be made to correspond
... 

 books.google.com is a U replacement system for some A bounded ontology U. Moreover such an
ontology U is unique up to isomorphism. Proof. We will need to use the
Representation Theorem for form systems whose statement and proof may be
found in the ... 

 books.google.com fine gradings of inner type, up to equivalence, 248 fine gradings of outer type, up
to equivalence, 259 gradings on F4, 196 gradings on G2, 146 gradings on Albert
algebra fine gradings up to equivalence, 184 up to isomorphism, 189 Weyl ... 

 books.google.com Every algebra with a sufficiently inclusive domain has a proper extension with the
same finitely generated subalgebras up to isomorphism. THEOREM 3.
Propositions 3 and 4 are provably equivalent to the existence of a measurable
cardinal, ... 

 books.google.com Definition V.1.2. The group Br(k) is called the Brauer group of k. Notice that by
Lemma V.1.1 (5), any class [A] ∈ Br(k) may be represented by a central division k
algebra D, which is unique up to isomorphism. Hence the map D−→ [D] yields a
... 

 books.google.com 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. How many Abelian groups (up to
isomorphism) are there a. of order 6? b. of order 15? c. of order 42? d. of order pq
, where p and q are distinct primes? e. of order pqr, where p, q, and r are distinct
primes? f. 

 books.google.com ... yield a partition of an unordered tree in bottomup subtree isomorphism
equivalence classes. While the isomorphism code for an unordered tree in
Definition 4.7 allows the reconstruction of a unique (up to isomorphism)
unordered tree, the. 

 books.google.com Theorem 2.15 enabled us to see that the passage from complex semisimple Lie
algebras to reduced abstract root systems is well defined up to isomorphism, and
we now want to see that it is oneone and onto, up to isomorphism. First we ... 

 books.google.com A central subject in Mathematics is classifying, that is characterizing the objects of
a given class up to equivalence relations, for instance the structures of a given
language, or the models of a given theory up to isomorphism. Let us mention ... 

 books.google.com D 7.9 Corollary The Zalgebraic ordered sets are, up to isomorphism, the Z ideal
extensions of Zadequate ordered sets. On the other hand, the Zadequate
ordered sets are precisely the Zspectra of Zalgebraic ordered sets. 7.10
Corollary ... 

 books.google.com Suppos_e that for some k with 0€k €a  1, 5,. contains a unique irreducible R[Ck]
module up to isomorphism. Then bk contains a unique irreducible R[C,.] module
up to isomorphism. Furthermore up to isomorphism b0 contains a unique ... 

 books.google.com Up to isomorphism in E8 there is a unique primitive embedding of D5 into E8
given by (e2, e5, e4, e3, e1) [18]. Moreover (Djfifs I (67, 63, 362 + 261 + 463 +
664 + 565 + 466 + 367 + 283) I A3 Thus we get three primitive embeddings of D5
Q A1 ... 

 books.google.com Second, the equational graphs are exactly the graphs of bounded treewidth (cf. [
17]) which are definable up to isomorphism by MSformulć (cf. [5]) according to
MSlogic. Note that these results generalise the fundamental results of [1] and [18
... 

 books.google.com In other words, for n > 3, to classify npointed smooth rational curves up to
isomorphism is the same as classifying ntuples of distinct points in a fixed P1, up
to projective equivalence, as in the Prologue. So the following result is just a ... 

 books.google.com Clearly, the dual fibration is just defined up to isomorphism since the choice of
the pseudofunctor P can only be made up to isomorphism. Let us conclude this
section with a point of terminology, related once more with the fact that the ... 

 books.google.com (a) Up to isomorphism, Z1 is the unique group oforder 1. (b) Up to isomorphism,
Z2 is the unique group oforder 2. (c) Up to isomorphism, Z3 is the unique group
oforder 3. (d) Up to isomorphism, there are exactly two groups of order 4, namely
... 

 books.google.com Then there are up to isomorphism only finitely many representations ofK with
restriction to Rad(K) isomorphic to E. Proof. By Lemma 1.2.2, we may assume
that k is algebraically closed. Since K0 is a quotient of Rad(K0)×(K0)der, and (K0)
der is ... 

 books.google.com mined, up to isomorphism, by the cardinality of the set of urelements and the least
ordinal not in the model.12 That is not much more than a restatement of
Zermelo's results discussed in §V.4. Recall that he also in effect showed that
every ... 

 books.google.com The summands E,A and T are uniquely determined up to isomorphism by the s
invariants of M. The summand C is likewise unique up to isomorphism. Proof. We
regress to (1.11) which states that M decomposes (uniquely up to isomorphism) ... 

 books.google.com Up to isomorphism P has a finite number of connected filters. 4. P has no ω
branch. In our proof of Theorem 6, we will make use of several observations. The
following fact depends on our assumption that pomsets are countable.
Proposition 2. 

 books.google.com Let X be a closed AutΛ (P)stable subvariety of GrassTd and C the class of
modules represented by the orbits of X. Then there is a fine (equivalently, a
coarse) moduli space classifying the modules in C up to isomorphism if and only
if C ... 

 books.google.com (a) The category mod(RQ) has exactly 3 indecomposable modules up to
isomorphism if and only if F — G. (b) The category mod(RG) has exactly 4
indecomposable modules up to isomorphism if and only if dim G F = dim pG = 2. (
c) The ... 

 books.google.com Then the front faces are pullbacks with (AG ,t )andthetopface commutes. Now the
van Kampen property in the opposite direction implies that the top face is a
pushout in GAGraphs.Thisimpliesthat(AG,t)and(AG ,t) are equal up to
isomorphism. 

 books.google.com Also, notice that for any C —> B, we have (f1,f2>°h=(f1°h,fz°h>Here is a neat way
to show that products are unique up to isomorphism. Given objects Al, A2 of C let
CAl,A2 be the category whose objects are pairs of Carrows Al <— B ;> A2 and ... 

 books.google.com (b) There is, up to isomorphism, only one simple Rmodule S, namely the column
vector module Dn×1. R acts faithfully on S and the left Rmodule R is isomorphic
to the direct sum Sn ofn copies of S. (c) EndRS ∼= D (if Rendomorphisms are ... 

 books.google.com (i) a complete theory that has more than one countable model up to isomorphism
must have a complete nonprincipal type; (ii) any complete type can be realized
in a countable model of its theory; (iii) a nonprincipal type of a complete theory ... 

 books.google.com We classify the complete firstorder theories of a linear ordering with a
distinguished subor dering, by examining the cardinalities in which the
distinguished part of a model determines the whole of the rest of the model, up to
isomorphism over ... 

 books.google.com PageRank values are unique up to isomorphism, i.e., vertices have the same
PageRank value across all input graph isomorphs. Hence, vertices contained in
the same block of the orbit partition also must have equal PageRank values. In
other ... 

 books.google.com Here, QA(M) denotes the first syzygy of M over A given by the projective cover of
M over A. We also show that if M in mod r is liftable to A and Ext^Af, M) = (0), then
the lifting is unique up to isomorphism. This is done by first showing that if M in ... 

 books.google.com F60: [EdRe97] Any two Catalan triangulations of the Mobius band with the same
number of vertices are equivalent under diagonal flips, up to isomorphism. F61: [
CoNaOOa] Any two Catalan triangulations of the punctured torus with the same ... 

 books.google.com (20) Show that, for every positive integer v > 2. there is a unique (up to
isomorphism) 2(v, 2, l) design. (2l) Show that any 2( v. 2. A) design is
isomorphic to the Afold multiple of a 2(u, 2, l) design. (22) Construct a 2(7, 3, 3)
design without ... 

 books.google.com ... class of locally isomorphic Lie groups there is up to isomorphism just one
simply connected Lie group such that every other member of the class is (up to
isomorphism) the quotient of this simply connected one by a discrete central
subgroup. 

 books.google.com For k > 1, let us define A(k) to be the unique (up to isomorphism) 2, 3, 4program
with the signature (0, fc,0), and C(k) and C'(k) to be the unique (up to
isomorphism) 2, 3, 4programs with the signatures (2, k — 1, 0) and (0, k — 1, 1),
... 

 books.google.com Introduction. At first glance torsion.free groups seem rather concrete and
accessible: up to isomorphism they are just the additive subgroups of rational
vector spaces. However, even the additive group Q of rational numbers itself
contains 2"° ... 

 books.google.com Let P, be an indecomposable, projective I\lattice for 1 < i < 2, and let P be the
pullback of PI and P2 mapping onto a simple Fmodule. By [Kl, Theorem 2.3], the
indecomposable Alattices are (up to isomorphism) PI, P2 and P. Thus, if I is a (
left) ... 

 books.google.com Theorem 15.1 (Shult and Thas [35]) Up to isomorphism the polar space Q+(5,q),
q odd, admits exactly one lsystem. For small parameters several results were
obtained, mostly with the aid of a computer. Here we just mention one recent ... 

 books.google.com (a) Show that up to isomorphism there are just countably many finitely generated
commutative rings with 1. [Use Hilbert's basis theorem.] (b) Show that there is no
countable universal commutative ring with 1 . [Let n be any set of primes, and ... 

 books.google.com A classification up to isomorphism of blockrigid crqgroups of ring type is
obtained. 1. Introduction Classification problem is always of great importance and
for blockrigid crqgroups it was solved up to nearisomorphism, see ... 

 books.google.com It is not difficult to see that, up to isomorphism, the isometry groups of 0
dimensional locally compact Polish metric spaces are exactly the closed
subgroups of S<x>. We do not know an exact characterization of the isometry
groups of pseudo ... 

 books.google.com This projective plane is usually denoted by PG(2,n) (PG stands for "projective
geometry," the 2 stands for "twodimensional"). The Fano plane is the projective
plane associated with the field of order 2. In fact, it is unique up to isomorphism
and ... 

 books.google.com 11.7 An application to the classification of principal Gbundles up to isomorphism
The notion of a horizontal lift can be used to say something about the set of
isomorphism classes of principal Gbundles over any given manifold. To this end
... 

 books.google.com Whenever char(fc) \ n, the moduli space An is defined to be the variety which
classifies up to isomorphism triples (X,C,<f>n), where X is an abelian variety, Ł is
a degree one, ample line bundle on X and <j>n is an isomorphism of Xn with ... 

 books.google.com An even more fundamental object in many respects is a moduli space, whose
points parametrize a set of varieties up to isomorphism. In a sense, the shift in
emphasis corresponds to the change in point of view between the last century
and this ... 

 books.google.com The tensor product M <S>r N, if it exists, is determined up to isomorphism by the
universal property. Proof. Let C be the following category. An object in C is an
ordered pair (P,p), where P is an .Rmodule and p : M x N — > P is a bilinear map
. 

 books.google.com By 11.14, K(d) is determined up to conjugation in H(d), so G is determined up to
isomorphism by Theorem D. Finally by 11.11 and 11.12, 0~+2 (^) satisfies our
hypothesis, so the lemma holds. (15.10) Let T be the set of transvections in U ... 

 books.google.com Recall that the trivial /cPmodule k is the unique simple fcPmodule up to
isomorphism and that the free module kP is the unique indecomposable
projective fcPmodule up to isomorphism (projective modules are free). However,
there are ... 

 books.google.com It turns out that for curves, T>(X) determines X (up to isomorphism) except in the
very special case when X is homeomorphic to the affine line A1 (we call such a
curve a framed curve). There are uncountably many nonisomorphic framed
curves ... 

 books.google.com Similarly, if a Lie group action preserves (L, V) up to isomorphism, a central
extension of the Lie group acts on sections of the line bundle (Kostant). A similar
central extension exists in a holomorphic context (Mumford). In §2.5, we discuss ... 

 books.google.com (2) Show that in 1.5.2 the assumption that A and B are of finite type can be
omitted. 1.5.4. Theorem. Let X and Y be two affine kvarieties. (i) A product variety
X x Y exists. It is unique up to isomorphism; ( ii) If X and Y are irreducible then so
is X ... 

 books.google.com 1. Introduction The classification of finite algebraic structures is an important task
in research in pure mathematics. Often, the first step toward full classification is to
determine how many structures exist up to isomorphism for each cardinality. 

 books.google.com We show <& is uniquely determined by (7.9) (7.12) up to isomorphism of
Leonard systems. Recall that <? is determined up to isomorphism of Leonard
systems by its own parameter array. We show the parameter array of <P is
determined by ... 

 books.google.com John Patrick Campbell Greenlees, Robert Ray Bruner, Nicholas John Kuhn  2001  Preview  More editions We say that X is formal if it is determined up to isomorphism by its homology. It is
not hard to check that X is formal if its homology has injective dimension 0 or 1 in
At(T). In general, if H(X) = (ty® V — > Q), the object X is equivalent to the fibre of
... 

 books.google.com In [125], they show that this class determines the *product up to isomorphism
and that it agrees with Fedosov's curvature for the ikproducts constructed by his
method. By Moser's classification [121] of nearby symplectic structures by their ... 

 books.google.com In the general formulation, a realization theorem for endomorphism rings states
that rings from this class can be represented up to isomorphism as
endomorphism rings of groups in a given class of groups. In such realization
theorems the ... 

 books.google.com I also include a proof ofthe fact that a right puresemisimple ring is right artinian
and that each of its indecomposable modules is of finite length. A ring is of finite
representation type if it is right puresemisimple and if there are, up to
isomorphism ... 

 books.google.com 19 that is, the classification of all topological spaces up to homeomorphism. If the
classification of objects up to isomorphism is the ultimate aim of a mathematical
theory, in practice we usually settle for something less—determining when two ... 

 books.google.com A comprehensive reference on combinatorial classification algorithms, with emphasis on both the general theory and application to central families of combinatorial objects, in particular, codes and designs. 

 books.google.com Two obvious senses of 'distinguish' here are 'up to isomorphism', and 'up to
elementary equivalence'. The sense used in [16] was the former. (In [4] it was
initially stated that this problem was still open, but then it was remarked in proof
that the ... 

 books.google.com ... almost all of which are unramified, such that 71: '1 ®1r,. The factors 7L], are
unique up to isomorphism. (2) For each isomorphism class 7 of continuous
irreducible representations of K. the Tisotypic component 0f71' has finite
dimension. 

 books.google.com ... which is an isomorphism of the uniform space E onto a dense uniform
subspace t(E) of E; moreover, E is determined up to isomorphism (as a uniform
space) by these requirements. We recall that one moreorless concrete
realization of E is ... 

 books.google.com If X is Witt, for example a complex algebraic variety, then SD(X) contains, up to
isomorphism, only one object 1C*, = 1C*. If X is not Witt, for example a real
algebraic variety whose links earn middle dimensional intersection homology,
then ... 

 books.google.com General schemes for classification up to isomorphism are contained in Krapf
Mutzbauer [KM], also Schultz [Sch85], and finally in Mader Vinsonhaler [MVl]. In
the present paper we adopt the approach and philosophy of the last paper and
give ... 

 books.google.com and hence H/II is a well defined Riemann surface (up to isomorphism). The
correspondence so indicated is welldefined. To see that it is bijective, start with
two isomorphic pelliptic curves f : Cl i> C2 and Zp actions S, : Zp >< C, I C,.
Consider ... 

 books.google.com Motivated by the study of approximately finitedimensional complex C*algebras
and period 2 antiautomorphisms and automorphisms thereon, two analogous
kinds of direct limit algebras are simultaneously classified (up to isomorphism) in
... 

 books.google.com Something to keep in mind while slogging through these definitions and technical
lemmas is the fact (see p.221 of [11]) that the simplicial sheaf cohomology ring H*
(Z, Z/2) in the representable case coincides up to isomorphism with the usual ... 

 books.google.com First, all embeddings, up to isomorphism, of a 2connected graph in the plane
can be obtained by a sequence of twists of blocks of the graph attached at two
vertices; and, second, any 3connccted graph has, up to isomorphism, at most
one ... 

 books.google.com This concept was introduced by Grothendieck with the term fibred category over
X. It is a generalization of the presheaf of categories where the key feature is that
transitivity of restriction is just defined up to isomorphism. Although the entire ... 

 books.google.com In this section we introduce the ideas of isomorphism and direct product. These
will then be used to describe, up to isomorphism, all groups of order no more
than 8. Informally, we regard two groups G and H as being isomorphic ( 'of the
same ... 

 books.google.com ... any pgroup of coclass r then G has a normal subgroup N, of order bounded by
a function of p and r, such that G/N has a certain welldefined structure. This falls
short of a classification up to isomorphism first by ignoring the subgroup N, and ... 

 books.google.com ... J(T) determines ET up to isomorphism. 5.10. Moduli of Riemann Surfaces. The
idea of moduli as numerical parameters of Riemann surfaces of the same
topological type stems from the following example. Example 1. Any Riemann
surface S ... 

 books.google.com The generation of discrete structures up to isomorphism is interesting for
theoretical as well as for practical purposes. Mathematicians want to look at and
analyse structures and, for example, the chemical industry uses mathematical
generators ... 

 books.google.com To see this, note that by Theorem 23.1 of [Fuc], any such group decomposes into
a direct sum of copies of Q and the Pr ̈ufer groups Z(p∞) = {z ∈ C∗ : (∃n)zpn =
1} and that they are classified up to isomorphism by the sequence consisting of ... 

 books.google.com In particular, the oneparameter specialization of a branch C to Cr which we have
seen above can be obtained in this way, up to isomorphism, by a map h\ : (C,0)
— y (Cr,0). The map p is equivariant with respect to the action of the group C* on
... 

 books.google.com But the realism to which we are entitled fails to be a complete vindication of
Frege's program because the basic laws of arithmetic derivable from Hume's
principle allow us to capture the natural numbers at most only up to isomorphism;
... 

 books.google.com Early theorems in the subject followed in the footsteps of other branches of
algebra by aiming to classify semigroups up to isomorphism. An early success in
this direction, and arguably the first theorem about finite semigroups, was the ... 

 books.google.com Call this the "existence condition." The second requirement is that at most one
structure (up to isomorphism) is described. This is the "uniqueness condition."
Uniqueness is less important, but let us start with that. Consider an implicit
definition ... 

 books.google.com In this paper the problem of L(2, 1)labeling unigraphs, i.e. graphs uniquely
determined by their own degree sequence up to isomorphism, is addressed and
a 3/2approximate algorithm for L(2, 1)labeling unigraphs is designed. This
algorithm ... 

 books.google.com As was noted in 5.2.3 and 5.3.2 the GQ of orders (2, 2) and (2, 4) are unique up to
isomorphism. Nevertheless it seems worthwhile to consider briefly an
independent construction for these two examples, the first of which was
apparently first ... 

 books.google.com Although Skolem's proof was specifically arithmetical, it was soon recognised that
the phenomenon was quite general: A firstorder theory could not hope to
characterise any structure up to isomorphism unless the structure were finite. This
is a ... 

 books.google.com ... among the strings obtained by concatenating the labels along the vertices and
edges of the cyclic permutations of C and its reverse. π is unique up to
isomorphism, hence it indeed provides a canonical string representation of
simple cycles. 

 books.google.com that every field K has an algebraic closure and moreover that algebraic closures
of K are unique up to isomorphism. Note, however, that these notions do not
serve to give us a well understood substitute for C. While C contains an
algebraic ... 

 books.google.com We characterize which legal strings obtain the same reduction graph (up to
isomorphism), and moreover we characterize which graphs are (isomorphic to)
reduction graphs. 1 Introduction Ciliates form a large group of onecellular
organisms ... 

 books.google.com For instance. while smooth plane conics are parametrized by a Zariski open
subset of P°, they are all isomorphic to P1. so up to isomorphism the parameter
space is a single point. Of course. the way to parametrize the possible degree
two ... 

 books.google.com In general a finite simple group G is not uniquely determined up to isomorphism
by a given centralizer H of a 2central involution z of G. The known examples
have already been mentioned in the introduction. It is the purpose of this chapter
to ... 

 books.google.com This implies that in order to classify semisimple Lie algebras up to isomorphism,
we only have to classify the possible Cartan subalgebras (which is trivial) and
root systems. It turns out that in fact the whole information about the root system $
is ... 

 books.google.com Leovigildo Alonso Tarrío, Ana Jeremías López, Joseph Lipman  1999  Preview But the dualizing objects we deal with are determined only up to isomorphism.
The Residue Theorems run deeper in that they include a canonical realization of
dualizing data, via differential forms. (See the above remarks on the HiiblKunz ... 

 books.google.com As a main syntactic theorem, we prove that the generic translation is involutive up
to isomorphism. As corollaries, we obtain full completeness results for the original
callbyvalue and call byname translations. The main syntactic theorem is ... 

 books.google.com If the field K contains the field K and if every polynomial P ∈ K[X] splits in K, then
the set K of roots in K of polynomials in K[X] is an algebraically closed field
containing K, and it is the smallest such field, unique up to isomorphism. One
calls K ... 

 books.google.com ... of / over a fixed field K. The category Mat/ is additive and each of its objects has
a decomposition into a finite direct sum of indecomposable objects, which is
unique up to isomorphism and permutation of indecomposable summands. 

 books.google.com This gives a bijection of the set of triples (E, p, a) (up to isomorphism) onto SLr(C(
(z))). To get rid of the the trivializations, we have to mod out by the automorphism
group of the trivial bundle over D and X — p. We get the following diagram: {E,p ... 

 books.google.com ... that E(R)–algebras are precisely the commutative generalized E(R)–algebras.
Every generalized E(R)–algebra is an E(R)– module and every E(R)–module
admits a generalized E(R)–algebra structure, which is unique up to isomorphism. 

 books.google.com The form E described above is a principal polarization of Jac(M). Let A = Jac(M)
and let ^ denote any principal polarization of A. Then, a theorem of Torelli asserts
that the pair (A, i/>) determines the curve represented by M up to isomorphism. 

 books.google.com In the unpublished manuscript [8], Kraft showed that, fixing g, there are only
finitely many such group schemes, up to isomorphism. (As we shall discuss later,
Kraft also gave a very nice description of all possible types.) About 20 years later,
this ... 

 books.google.com This notion is based on the notion of symmetric net category of [22] (Section 7.6);
these are categories with two multiplications, ∧ and ∨, associative and
commutative up to isomorphism, which have moreover arrows of the dissociativity
type A ... 

 books.google.com As in the real case, the above condition can be replaced by the seemingly
stronger condition analogous to (2.2). As always, the universal property
determines C(V) up to isomorphism, and the construction is C(V) = T(V)/I where T
(V) is the ... 

 books.google.com The complement of a knot is uniquely specified (up to isomorphism) by its
peripheral group system which consists of the fundamental group and a few
subgroups thereof (this is Waldhausen's theorem [73], see [40] for a more
accessible proof). 

 books.google.com Actually, Hurwitz numbers depend on a Riemann surface P up to isomorphism, a
number of critical values and a set of partitions on assigned to points of P that are
critical values of /. Clearly, the partitions of n are in onetoone correspondence ... 

 books.google.com ... one orthogonal group (up to isomorphism) in this case, and we write it as GO(V
)orGOn(q) without ambiguity. If n is even, however, we have two thoroughly
different orthogonal groups (they do not even have the same order, as we shall
see). 

 