We prove that endomorphisms of adjoint Chevalley groups are inner, indicating Sha-rigidity in group theory.
We prove that every locally inner (class-preserving) endomorphism of adjoint Chevalley groups and their elementary subgroups over commutative rings is inner for the root systems A1, A2, B2 (assuming 2 is invertible in the ring), and for G2 (assuming 2 and 3 are invertible). As a consequence, these groups are Sha-rigid. The proofs are direct and do not rely on classification of automorphisms or structural results about injective endomorphisms. Published in the journal of Groups, Complexity, Cryptology
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Bunina et al. (2026) studied this question.
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