automorphism love

# automorphism

## Definitions

• n. An isomorphism of a mathematical object or system of objects onto itself.

### from the GNU version of the Collaborative International Dictionary of English

• n. Automorphic characterization.

### from The Century Dictionary and Cyclopedia

• n. The ascription of one's own characteristics to another, or the habit of judging others or explaining their acts by means of analogies furnished by the knowledge of one's self.
• n. In mathematics, the property of remaining unchanged by all the substitutions of any group of linear transformations.
• n. Anthropomorphism; the conception of the lower animals by analogy with man.

## Etymologies

auto- +‎ morphism

## Examples

• This has as its automorphism groups F_4 whose root space is a minimal tessellation in 4 dimensions and G_2.

Einstein Still Rules, Says Fermi Telescope Team | Universe Today

• But since Ï is an automorphism, it also preserves function application, so that Ï (f (U)) = Ïf (Ï (U)).

The Axiom of Choice

• Sym (V) is also a model of set theory with set of atoms A, and Ï induces an automorphism of Sym (V).

The Axiom of Choice

• If is a model of Z (Zermelo set theory) with an external automorphism Ï and an ordinal Îº such that Îº Ï (Îº), then

Quine's New Foundations

• If the space is a so-called metric space, it is usually expected that the underlying automorphism shall at least be isometric, meaning that the transformations preserve the distance between any pair of points.

SYMMETRY AND ASYMMETRY

• Obviously the homogeneity of a space is the more interesting if the underlying group of automorphism is more important.

SYMMETRY AND ASYMMETRY

• In general, an automorphism of a space, any space, is an invertible transformation of the entire space into itself, and such a transformation is nothing other than a rearrangement of the totality of the points of the space from their given ordering into any other.

SYMMETRY AND ASYMMETRY

• Inasmuch as a mirror reflection of the space is a transformation of the entire space into itself it is called an automorphism.

SYMMETRY AND ASYMMETRY

• If such a group is given and held fast, then a figure in space is called symmetric if each automorphism of the group leaves it unchanged.

SYMMETRY AND ASYMMETRY

• Moore determinant: A determinant defined over a finite field which has successive powers of the Frobenius automorphism applied to the first column.

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