In mathematics, an inner form of an algebraic group
over a field
is another algebraic group
such that there exists an isomorphism
between
and
defined over
(this means that
is a
-form of
) and in addition, for every Galois automorphism
the automorphism
is an inner automorphism of
(i.e. conjugation by an element of
).
Through the correspondence between
-forms and the Galois cohomology
this means that
is associated to an element of the subset
where
is the subgroup of inner automorphisms of
.
Being inner forms of each other is an equivalence relation on the set of
-forms of a given algebraic group.
A form which is not inner is called an outer form. In practice, to check whether a group is an inner or outer form one looks at the action of the Galois group
on the Dynkin diagram of
(induced by its action on
, which preserves any torus and hence acts on the roots). Two groups are inner forms of each other if and only if the actions they define are the same.
For example, the
-forms of
are itself and the unitary groups
and
. The latter two are outer forms of
, and they are inner forms of each other.