In mathematics, an inner form of an algebraic group
over a field
is another algebraic group
such that there exists an isomorphism
) and in addition, for every Galois automorphism
the automorphism
σ
(i.e. conjugation by an element of
-forms and the Galois cohomology
{\displaystyle H^{1}(\mathrm {Gal} ({\overline {K}}/K),\mathrm {Aut} (G))}
is associated to an element of the subset
{\displaystyle H^{1}(\mathrm {Gal} ({\overline {K}}/K),\mathrm {Inn} (G))}
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.
are itself and the unitary groups
The latter two are outer forms of
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