• 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...
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  • 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...
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  • 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...
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  • 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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  • ^{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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  • 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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  • 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...
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  • non-trivial automorphisms are outer automorphisms. Non-abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms. Group...
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  • 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...
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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...
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  • theory, a power automorphism of a group is an automorphism that takes each subgroup of the group to within itself. The power automorphism of an infinite...
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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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  • 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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  • 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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  • for every Galois automorphism σ ∈ G a l ( K ¯ / K ) {\displaystyle \sigma \in \mathrm {Gal} ({\overline {K}}/K)} the automorphism ϕ − 1 ∘ ϕ σ {\displaystyle...
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  • obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism. Let f : R → S be a ring homomorphism. Then, directly from these definitions...
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  • 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...
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  • 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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  • 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...
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  • subgroup of the automorphism group. Every inner automorphism is an IA automorphism. Torelli group Bachmuth, S. (1966), "Induced automorphisms of free groups...
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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...
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  • {\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...
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  • quotientable automorphism of a group is an automorphism that takes every normal subgroup to within itself. As a result, it gives a corresponding automorphism for...
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  • algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative...
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  • 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...
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  • of group theory, a group is said to be capable if it occurs as the inner automorphism group of some group. These groups were first studied by Reinhold Baer...
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  • 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...
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  • In mathematics, Out(Fn) is the outer automorphism group of a free group on n generators. These groups are at universal stage in geometric group theory...
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