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More precisely what is the name for the group

$$\{ X\mapsto \alpha^2X+\beta : \alpha,\beta \in GF(q), \alpha \neq 0\}$$ I've always called it the special affine group, but I see that can mean something else.

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    The map $X \mapsto \alpha^2 X$ maps quadratic residues to quadratic residues and nonresidues to nonresidues and thus is a restricted form of a general linear map.2012-08-02

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