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...

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

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...

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...

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...

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

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

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, 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...

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...

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...

^{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...

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...

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...

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...

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...

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

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...

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...

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...

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...