• In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called...
    11 KB (1,689 words) - 19:57, 18 April 2025
  • Thumbnail for Automorphism
    nontrivial automorphism: negation. Considered as a ring, however, it has only the trivial automorphism. Generally speaking, negation is an automorphism of any...
    11 KB (1,346 words) - 10:42, 24 February 2025
  • to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified;...
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  • Thumbnail for Dihedral group
    automorphism interchanging the two types of reflections (properly, a class of outer automorphisms, which are all conjugate by an inner automorphism)...
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  • 1982). Note that as an outer automorphism, it is a class of automorphisms, well-determined only up to an inner automorphism, hence there is not a natural...
    22 KB (3,213 words) - 17:25, 20 December 2024
  • automorphism φ of G, one has φ(H) ≤ H; then write H char G. It would be equivalent to require the stronger condition φ(H) = H for every automorphism φ...
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  • up to inner automorphisms. Automorphism groups appear very naturally in category theory. If X is an object in a category, then the automorphism group...
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  • the map f : G → Aut(G), from G to the automorphism group of G defined by f(g) = ϕg, where ϕg is the automorphism of G defined by f(g)(h) = ϕg(h) = ghg−1...
    12 KB (1,189 words) - 19:22, 28 May 2025
  • Thumbnail for Alternating group
    the automorphism group of An is the symmetric group Sn, with inner automorphism group An and outer automorphism group Z2; the outer automorphism comes...
    17 KB (1,539 words) - 05:01, 21 October 2024
  • ^{0}({\mathfrak {g}})} is known as the outer automorphism group. It is known that the outer automorphism group for a simple Lie algebra g {\displaystyle...
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  • its inner automorphism group Inn(G) (its quotient by its center) is simple (and it follows Inn(G) must be non-abelian simple, as inner automorphism groups...
    2 KB (296 words) - 23:43, 12 August 2023
  • obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism. Let f : R → S be a ring homomorphism. Then, directly from these definitions...
    12 KB (1,641 words) - 12:34, 6 May 2025
  • non-trivial automorphisms are outer automorphisms. Non-abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms. Group...
    12 KB (2,043 words) - 06:23, 21 December 2024
  • subgroup of the automorphism group. Some facts: Every inner automorphism is a class automorphism. Every class automorphism is a family automorphism and a quotientable...
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  • Thumbnail for Absolute Galois group
    group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism. It is a profinite...
    8 KB (950 words) - 03:00, 17 March 2025
  • Thumbnail for Symmetric group
    maps up to inner automorphism) corresponding to the outer automorphism of S6. S5 → S6 as a transitive subgroup, yielding the outer automorphism of S6 as...
    46 KB (6,212 words) - 15:23, 13 February 2025
  • Thumbnail for Homeomorphism
    right translation y ↦ y x , {\displaystyle y\mapsto yx,} and the inner automorphism y ↦ x y x − 1 {\displaystyle y\mapsto xyx^{-1}} are homeomorphisms...
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  • theory, the automorphism group of a free group is a discrete group of automorphisms of a free group. The quotient by the inner automorphisms is the outer...
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  • algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative...
    22 KB (3,091 words) - 12:09, 26 March 2025
  • Thumbnail for Homogeneous space
    of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group...
    15 KB (1,825 words) - 02:56, 3 May 2025
  • {\displaystyle R\to R,x\mapsto uxu^{-1}} is a ring homomorphism, called an inner automorphism of R. Let R be a commutative ring of prime characteristic p. Then...
    99 KB (13,738 words) - 11:06, 29 May 2025
  • its automorphism group. A complete group is fully normalized in any bigger group in which it is embedded because every automorphism of it is inner. Every...
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  • another graph In group theory, various notions are called conjugation: Inner automorphism, a type of conjugation homomorphism Conjugacy class in group theory...
    3 KB (407 words) - 14:29, 14 December 2024
  • F(1) induces the trivial automorphism. If on the other hand F is a quasiconformal lift of f inducing an inner automorphism of Γ, after composition with...
    43 KB (6,002 words) - 10:21, 3 May 2025
  • Thumbnail for P-group
    every finite p-group has a non-trivial outer automorphism group. Every automorphism of G induces an automorphism on G/Φ(G), where Φ(G) is the Frattini subgroup...
    21 KB (2,765 words) - 16:44, 24 May 2025
  • given class of outer automorphisms, it acts on the characters – because inner automorphisms act trivially, the action of the automorphism group A u t {\displaystyle...
    29 KB (4,615 words) - 03:49, 10 May 2025
  • algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative...
    6 KB (774 words) - 00:21, 24 September 2024
  • does act on itself as inner automorphisms. A group is said to be quasi-nilpotent if every element acts as an inner automorphism on every chief factor...
    9 KB (1,318 words) - 01:46, 6 September 2022
  • of Switzerland), a district in Switzerland named for the river Inn Inner automorphism, in mathematics, a set of functions on groups Inns of Court, London...
    690 bytes (124 words) - 06:28, 22 March 2025
  • g(a) = b · f(a) · b−1. In particular, every automorphism of a central simple k-algebra is an inner automorphism. First suppose B = M n ⁡ ( k ) = End k ⁡...
    4 KB (683 words) - 04:50, 25 January 2024