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In mathematics, an inner form of an algebraic group G {\displaystyle G} over a field K {\displaystyle K} is another algebraic group H {\displaystyle H} such that there exists an isomorphism ϕ {\displaystyle \phi } between G {\displaystyle G} and H {\displaystyle H} defined over K ¯ {\displaystyle {\overline {K}}} and in addition, for every Galois automorphism σ ∈ G a l {\displaystyle \sigma \in \mathrm {Gal} } the automorphism ϕ − 1 ∘ ϕ σ {\displaystyle \phi ^{-1}\circ \phi ^{\sigma }} is an inner automorphism of G {\displaystyle G} {\displaystyle G} ].
Through the correspondance between K {\displaystyle K} -forms and the Galois cohomology H 1 , I n n ] {\displaystyle H^{1},\mathrm {Inn} ]} this means that H {\displaystyle H} is associated to an element of the subset H 1 , I n n ] {\displaystyle H^{1},\mathrm {Inn} ]} where I n n {\displaystyle \mathrm {Inn} } is the subgroup of inner automorphisms of G {\displaystyle G}.
Being inner forms of each other is an equivalence relation on the set of K {\displaystyle K} -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 G a l {\displaystyle \mathrm {Gal} } on the Dynkin diagram of G {\displaystyle G} {\displaystyle G} , 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.