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Properties of Affine Groups over Finite Fields

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Let F be a field. The 1-dimensional affine group over F is the set A of all functions f: F -> F of the form f(x) = ax+b where a, b belong to F and a is a unit. Let S and T be subsets of A consisting of the scaling (s(x)=ax) and translations (t(x) = x+b)

Are S and T normal subgroups?
If p > 3 and F = Z/p consider the subgroup of D of A generated by T and s(x) = -x. What is the order |D| and can you find an isomorphism of D with a known group?
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Solution Summary

We investigate properties of two subgroups of the group of affine transformations over Z/pZ, namely the group of translations and the group of scale transformations.

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Let F be a field. The 1-dimensional affine group over F is the set A of all functions f: F  F of the form f(x) = ax+b where a, b belong to F and a is a unit. Let S and T be subsets of A consisting of the scaling (s(x)=ax) and translations (t(x) = x+b)

Are S and T normal subgroups?
We claim that T is a normal subgroup but S is not. To see ...

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