Inner automorphism

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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.

Definition[edit]

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

is called (right) conjugation by g (see also conjugacy class). This function is a homomorphism of G: for all ,

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.[1]

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[edit]

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).

Inn(G) is a normal subgroup of the full automorphism group Aut(G) of G. The outer automorphism group, Out(G) is the quotient group

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 = xax = xa.

Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group (or ring).

An automorphism of a group G is inner if and only if it extends to every group containing G.[2]

By associating the element aG 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[edit]

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:

  1. G is nilpotent of class 2
  2. G is a regular p-group
  3. G/Z(G) is a powerful p-group
  4. The centralizer in G, CG, of the center, Z, of the Frattini subgroup, Φ, of G, CGZ∘Φ(G), is not equal to Φ(G)

Types of groups[edit]

The inner automorphism group of a group G, Inn(G), is trivial (i.e., consists only of the identity element) if and only if G is abelian.

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.

If the inner automorphism group of a perfect group G is simple, then G is called quasisimple.

Lie algebra case[edit]

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.

Extension[edit]

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.

References[edit]

  1. ^ S., Dummit, David (2004). Abstract algebra. Foote, Richard M., 1950- (3. ed.). Hoboken, NJ: Wiley. p. 45. ISBN 9780471452348. OCLC 248917264.
  2. ^ Schupp, Paul E. (1987), "A characterization of inner automorphisms" (PDF), Proceedings of the American Mathematical Society, American Mathematical Society, 101 (2): 226–228, doi:10.2307/2045986, JSTOR 2045986, MR 0902532

Further reading[edit]