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 the...

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

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;...

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...

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

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

^{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...

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...

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...

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...

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...

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

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...

automorphisms, which may be identified with the automorphisms of the group of permutations on six elements (Coxeter 1958b). The inner automorphisms of...

subgroup of the automorphism group. Some facts: Every inner automorphism is a class automorphism. Every class automorphism is a family automorphism and a quotientable...

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...

subgroup of the automorphism group. Every inner automorphism is an IA automorphism. Torelli group Bachmuth, S. (1966), "Induced automorphisms of free groups...

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...

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...

p) is the outer automorphism group of the group Zpn, and also the automorphism group, because Zpn is abelian, so the inner automorphism group is trivial...

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

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...

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

power automorphism of a group is an automorphism that takes each subgroup of the group to within itself. It is worth noting that the power automorphism of...

another graph In group theory, various notions are called conjugation: Inner automorphism, a type of conjugation homomorphism Conjugation in group theory, related...

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

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