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...

nontrivial automorphism: negation. Considered as a ring, however, it has only the trivial automorphism. Generally speaking, negation is an automorphism of any...

to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified;...

automorphism interchanging the two types of reflections (properly, a class of outer automorphisms, which are all conjugate by an inner automorphism)...

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...

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...

up to inner automorphisms. Automorphism groups appear very naturally in category theory. If X is an object in a category, then the automorphism group...

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...

the automorphism group of An is the symmetric group Sn, with inner automorphism group An and outer automorphism group Z2; the outer automorphism comes...

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...

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...

non-trivial automorphisms are outer automorphisms. Non-abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms. Group...

obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism. Let f : R → S be a ring homomorphism. Then, directly from these definitions...

^{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...

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...

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...

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...

algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative...

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...

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...

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...

{\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...

algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative...

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...

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...

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...

algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative...

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...

determined by an outer automorphism. If two automorphisms φ , ψ {\displaystyle \varphi ,\psi } represent the same outer automorphism, that is, φ = ψ ι {\displaystyle...

algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative...