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# isomorphism

[ ahy-s*uh*-**mawr**-fiz-*uh*m ]

## noun

- the state or property of being isomorphous or isomorphic.
*Mathematics.*a one-to-one relation onto the map between two sets, which preserves the relations existing between elements in its domain.

isomorphism

/ ˌaɪsəʊˈmɔːfɪzəm /

## noun

- biology similarity of form, as in different generations of the same life cycle
- chem the existence of two or more substances of different composition in a similar crystalline form
- maths a one-to-one correspondence between the elements of two or more sets, such as those of Arabic and Roman numerals, and between the sums or products of the elements of one of these sets and those of the equivalent elements of the other set or sets

isomorphism

/ ī′sə-môr**′**fĭz′əm /

- Similarity in form, as in organisms of different ancestry.
- A one-to-one correspondence between the elements of two sets such that the result of an operation on elements of one set corresponds to the result of the analogous operation on their images in the other set.
- A close similarity in the crystalline structure of two or more substances of different chemical composition. Isomorphism is seen, for example, in the group of minerals known as garnets, which can vary in chemical composition but always have the same crystal structure.

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## Word History and Origins

Origin of isomorphism^{1}

First recorded in 1820–30; isomorph(ous) + -ism

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## Example Sentences

The following table shows where isomorphism may be generally expected.

From Project Gutenberg

Identity, or approximate identity, of crystal form is not in itself sufficient to establish true isomorphism.

From Project Gutenberg

The substitution on the symbols of the operations of a group resulting from such a correspondence is called an outer isomorphism.

From Project Gutenberg

A subgroup of a group G, which is transformed into itself by every isomorphism of G, is called a characteristic subgroup.

From Project Gutenberg

Moreover, the isomorphism is simple unless for one or more operations, other than identity, the sets all remain unaltered.

From Project Gutenberg

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