Isomorphism (3587 bytes)
1: In [[mathematics]], an '''isomorphism''' (in [[Greek language|Greek]] ''isos'' = equal ...
3: :The word "isomorphism" applies when two complex structures can be mappe...
5: Formally, an isomorphism is a [[bijective]] map ''f'' such that both ''f''...
7: If there exists an isomorphism between two structures, we call the two structure...
13: ...Y'' with an ordering <nowiki>[=,</nowiki> then an isomorphism from ''X'' to ''Y'' is a bijective function ''f''...

Order isomorphism (1354 bytes)
1: ... Two strictly weaker notions that relate to order isomorphisms are [[order embedding]]s and [[Galois connection...
3: ...red sets (''S'', &le;) and (''T'', <=) an ''order isomorphism'' from (''S'', &le;) to (''T'', <=) is a [[surjec...
5: ...e]]. Hence, yet another characterization of order isomorphisms is possible: they are exactly those monotone [[b...
7: An order isomorphism from (''S'', &le;) to itself is called an [[order...

Isomorphism of categories (4523 bytes)
5: Isomorphism of categories is a very strong condition and rare...
9: As is true for any notion of [[isomorphism]], we have the following general properties forma...
14: A functor ''F'' : ''C'' &rarr; ''D'' yields an isomorphism of categories if and only if it is [[bijective]] ...
18: ...ule (mathematics)|left module]]s over ''kG''. The isomorphism can be described as follows: given a group repres...
25: Another isomorphism of categories arises in the theory of [[Boolean a...

Ring isomorphism (271 bytes)
1: In [[abstract algebra]], a '''ring isomorphism''' is a [[bijective]] [[ring homomorphism]] (a [[...

Uniform isomorphism (677 bytes)
1: ...''' or '''uniform homeomorphism''' is a special [[isomorphism]] between [[uniform spaces]] which respects [[uni...
5: ...iform spaces ''X'' and ''Y'' is called '''uniform isomorphism''' if it satisfies the following properties
12: *[[homeomorphism]] is an isomorphism between [[topological space|topological spaces]]

Group isomorphism (3064 bytes)
1: ...ts the given group operations. If there exists an isomorphism between two groups, then the groups are called ''...
5: ...'' to ''H''. Spelled out, this means that a group isomorphism is a bijective function ''f'' : ''G'' <tt>-></tt>...
10: ...tiplication ('''R'''⁺,&times;) via the isomorphism
14: ...]]s of absolute value 1 (with multiplication); an isomorphism is given by
20: ... last two examples, one cannot construct concrete isomorphisms; the proofs rely on the [[axiom of choice]].

Isomorphism class (528 bytes)
3: An '''isomorphism class''' is a collection of mathematical objects ...
6: Isomorphism classes are often defined if the exact identity o...

Isomorphism theorem (2784 bytes)
1: ...bra]], stating the existence of certain [[natural isomorphism]]s.
5: First we state the isomorphism theorems for [[group (mathematics)|groups]], wher...
7: === First isomorphism theorem===
14: === Second isomorphism theorem ===
18: === Third isomorphism theorem ===

Subgraph isomorphism problem (652 bytes)
1: ...omplexity theory|complexity theory]], '''Subgraph-Isomorphism''' is a [[decision problem]] that is known to be ...
3: Subgraph-Isomorphism(G₁, G₂)<br/>

Coimage (525 bytes)
12: ...ics)|image]] by the [[isomorphism theorem | first isomorphism theorem]], when that theorem applies.

Short five lemma (558 bytes)
4: ...i>h</i> are [[isomorphism]]s, then <i>f</i> is an isomorphism as well.

Commutative diagram (993 bytes)
5: For example, the [[isomorphism theorem|first isomorphism theorem]] is a commutative triangle as follows:

Splitting field (967 bytes)
1: ...p to]] isomorphism; the amount of freedom in that isomorphism is known to be the [[Galois group]] of P (if we a...

Bicategory (2030 bytes)
1: ...y]] used to extend the notion of sameness (i.e. [[isomorphism]]) to the [[morphism]]s of a category. A bicatego...
4: ...laxed from the usual equality to holding up to an isomorphism. The two types of composition (horizontal and ver...
6: *For 0-cells W,X,Y,Z, [[natural isomorphism]]s (with '''Id''' denoting the identity functor a...

Monomorphism (1535 bytes)
13: ...s a first factor, unless that epimorphism is an [[isomorphism]].
20: *[[isomorphism]]

Inner automorphism (1538 bytes)
8: ...G'') as above, one obtains an [[group isomorphism|isomorphism]] between the [[factor group]] ''G''/Z(''G'') (wh...

List of small groups (5465 bytes)
1: ... [[order (group theory)|order]] [[up to]] [[group isomorphism]].
12: ... prime.) We use the equality sign ("=") to denote isomorphism.
189: .... The groups are listed up to [[group isomorphism|isomorphism]]. At present, the library contains the following...

P-group (2841 bytes)
5: ...''-groups of the same order are not necessarily [[isomorphism | isomorphic]]; for example, the [[cyclic group]]...
10: a list of all the [[isomorphism class|isomorphism classes]]

Up to (2492 bytes)
7: ...sider groups to be equivalent if they are [[group isomorphism|isomorphic]].
9: ...simply "mod") for the same purpose, as in "modulo isomorphism, there are two groups of order 4", or "there are ...

Characterization (mathematics) (1771 bytes)
1: ...n of ''Q'' is a single [[equivalence class]] (for isomorphism, in the given example &mdash; depending on how ''...

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