automorphism wikipedia - EAS

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  1. Automorphism - Wikipedia

    https://en.wikipedia.org/wiki/Automorphism

    WebIn mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group.It is, loosely speaking, the symmetry group of the object.

  2. Monster group - Wikipedia

    https://en.wikipedia.org/wiki/Monster_group

    WebThe Schur multiplier and the outer automorphism group of the monster are both trivial. Representations. The minimal degree of a faithful complex representation is 47 × 59 × 71 = 196,883, hence is the product of the three largest prime divisors of the order of M. The smallest faithful linear representation over any field has dimension 196,882 ...

  3. Ring homomorphism - Wikipedia

    https://en.wikipedia.org/wiki/Ring_homomorphism

    WebEndomorphisms, isomorphisms, and automorphisms. A ring endomorphism is a ring homomorphism from a ring to itself.; A ring isomorphism is a ring homomorphism having a 2-sided inverse that is also a ring homomorphism. One can prove that a ring homomorphism is an isomorphism if and only if it is bijective as a function on the …

  4. Commutator subgroup - Wikipedia

    https://en.wikipedia.org/wiki/Commutator_subgroup

    WebIn mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.. The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian.In other words, / is …

  5. Dihedral group - Wikipedia

    https://en.wikipedia.org/wiki/Dihedral_group

    WebIn mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.. The notation for the dihedral group differs in geometry and abstract algebra.In geometry, D n …

  6. Alternating group - Wikipedia

    https://en.wikipedia.org/wiki/Alternating_group

    WebFor n = 3 the automorphism group is Z 2, with trivial inner automorphism group and outer automorphism group Z 2. The outer automorphism group of A 6 is the Klein four-group V = Z 2 × Z 2, and is related to the outer automorphism of S 6. The extra outer automorphism in A 6 swaps the 3-cycles (like (123)) with elements of shape 3 2 (like …

  7. Graph isomorphism - Wikipedia

    https://en.wikipedia.org/wiki/Graph_isomorphism

    WebThe Whitney graph isomorphism theorem, shown by Hassler Whitney, states that two connected graphs are isomorphic if and only if their line graphs are isomorphic, with a single exception: K 3, the complete graph on three vertices, and the complete bipartite graph K 1,3, which are not isomorphic but both have K 3 as their line graph. The Whitney graph …

  8. Gamma matrices - Wikipedia

    https://en.wikipedia.org/wiki/Gamma_matrices

    WebIn natural units, the Dirac equation may be written as =where is a Dirac spinor.. Switching to Feynman notation, the Dirac equation is (/) =The fifth "gamma" matrix, γ 5 It is useful to define a product of the four gamma matrices as =, so that = (in the Dirac basis). Although uses the letter gamma, it is not one of the gamma matrices of Cl 1,3 ().The number 5 is …

  9. Ring theory - Wikipedia

    https://en.wikipedia.org/wiki/Ring_theory

    WebIn algebra, ring theory is the study of rings —algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers.Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal …

  10. Morphism - Wikipedia

    https://en.wikipedia.org/wiki/Morphism

    WebIn mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in …



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