Everything about Inner Automorphism totally explained
In
abstract algebra, an
inner automorphism of a
group G is a
function » f :
G →
G
defined by
» f(
x) =
axa−1, for all
x in
G,
where
a is a given fixed element of
G.
The operation
axa−1 is called
conjugation (see also
conjugacy class). Informally, in a conjugation a certain operation is applied, then another one (
x) is carried out, and then the initial operation is reversed. Sometimes this matters ('take off shoes, take off socks, replace shoes'), and sometimes ('take off left glove, take off right glove, replace left glove') it doesn't.
In fact
» axa−1 =
x
is equivalent to saying
» ax =
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. This is one good reason to study this concept in group theory.
Notation
The expression
axa−1 is often denoted exponentially by
ax. This notation is used because we've the rule
a(
bx)=
abx (giving a left
action of
G on itself). An alternative form, leading to a right action, can be obtained by denoting
a−1xa as
xa.
Properties
Every inner automorphism is indeed an
automorphism of the group
G, for example it's a
bijective map from
G to
G and it's a
homomorphism (meaning
a(
xy) =
axay).
The inner and outer automorphism groups
The
composition of two inner automorphisms is again an inner automorphism (as mentioned above:
a(
bx)=
abx), and with this operation, the collection of all inner automorphisms of
G is itself a group, the inner automorphism group of
G denoted Inn(
G).
An automorphism of
G which isn't inner is called an
outer automorphism.
Inn(
G) is a
normal subgroup of the full
automorphism group Aut(
G) of
G. The
quotient group
» Aut(
G)/Inn(
G)
is known as the
outer automorphism group Out(
G). The outer automorphism group measures, in a sense, how many automorphisms of
G are not inner. Note however that the elements of Out(
G) are not the outer automorphisms (which don't form a group) but are
cosets of automorphisms. Every outer automorphism yields a non-trivial element of Out(
G), but different outer automorphisms may yield the same element of Out(
G).
By associating the element
a in
G with the inner automorphism
f(
x) =
ax in Inn(
G) as above, one obtains an
isomorphism between the
quotient group G/Z(
G) (where Z(
G) is the
center 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).
Types of groups
It follows that the group Inn(
G) of inner automorphisms is itself trivial (for example consists only of the
identity element)
if and only if G is
abelian.
Inn(
G) can only be a
cyclic group when it's trivial, by a basic result on the center of a group.
At the opposite end of the spectrum, it's possible that the inner automorphisms exhaust the entire automorphism group; a group whose automorphisms are all inner is called
complete.
If the inner automorphism group of a
perfect group G is simple, then
G is called
quasisimple.
Ring case
Given a
ring R and a
unit u in
R, the map
f(
x) =
uxu-1 is a
ring automorphism of
R. The ring automorphisms of this form are called
inner automorphisms of
R. They form a normal subgroup of the automorphism group of
R.
Lie algebra case
An automorphism of a
Lie algebra . 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
If
G arises as the
group of units of a
ring A, then an inner automorphism on
G can be extended to a
projectivity on the
projective space over
A by
inversive ring geometry. In particular, the inner automorphisms of the
classical linear groups can be so extended.
Further Information
Get more info on 'Inner Automorphism'.
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