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...

^{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...

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

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...

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

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...

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...

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

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...

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...

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...

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

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...

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...

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...

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...

{\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...

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...

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

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...

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...

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...

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...