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 conjugating element. These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup gives rise to the concept of the outer automorphism group.
If G is a group (or a ring) and g is an element of G (if G is a ring, then g must be a unit), then the function
where the third equality is given by the insertion of the identity between and . Furthermore, it has a left and right inverse, namely . Thus, is bijective, and so an isomorphism of G onto itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.
When discussing right conjugation, the expression is often denoted exponentially by . This notation is used because composition of conjugations is associative: for all . This shows that conjugation gives a right action of G on itself.
Inner and outer automorphism groups
The composition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of G is a group, the inner automorphism group of G denoted Inn(G).
- Out(G) ≡ Aut(G)/Inn(G)
The outer automorphism group measures, in a sense, how many automorphisms of G are not inner. Every non-inner automorphism yields a non-trivial element of Out(G), but different non-inner automorphisms may yield the same element of Out(G).
Saying that conjugation of x by a leaves x unchanged is equivalent to saying that a and x commute:
- a−1xa = x ⇔ ax = xa.
An automorphism of a group G is inner if and only if it extends to every group containing G.
By associating the element a ∈ G with the inner automorphism f(x) = xa in Inn(G) as above, one obtains an isomorphism between the quotient group G/Z(G) (where Z(G) is the centre of G) and the inner automorphism group:
- G/Z(G) = Inn(G).
This is a consequence of the first isomorphism theorem, because Z(G) is precisely the set of those elements of G that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).
Non-inner automorphisms of finite p-groups
A result of Wolfgang Gaschütz says that if G is a finite non-abelian p-group, then G has an automorphism of p-power order which is not inner.
It is an open problem whether every non-abelian p-group G has an automorphism of order p. The latter question has positive answer whenever G has one of the following conditions:
- G is nilpotent of class 2
- G is a regular p-group
- G/Z(G) is a powerful p-group
- The centralizer in G, CG, of the center, Z, of the Frattini subgroup, Φ, of G, CG∘Z∘Φ(G), is not equal to Φ(G)
Types of groups
The group Inn(G) is cyclic only when it is trivial.
At the opposite end of the spectrum, the inner automorphisms may exhaust the entire automorphism group; a group whose automorphisms are all inner and whose centre is trivial is called complete. This is the case for all of the symmetric groups on n elements when n is not 2 or 6, when n = 6 the symmetric group has a unique non-trivial class of outer automorphisms, and when n = 2 the symmetric group, despite having no outer automorphisms, is abelian, giving a non-trivial centre disqualifying it from being complete.
Lie algebra case
An automorphism of a Lie algebra 𝔊 is called an inner automorphism if it is of the form Adg, where Ad is the adjoint map and g is an element of a Lie group whose Lie algebra is 𝔊. The notion of inner automorphism for Lie algebras is compatible with the notion for groups in the sense that an inner automorphism of a Lie group induces a unique inner automorphism of the corresponding Lie algebra.
If G is the group of units of a ring, A, then an inner automorphism on G can be extended to a mapping on the projective line over A by the group of units of the matrix ring, M2(A). In particular, the inner automorphisms of the classical groups can be extended in that way.
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