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Let F=GF(p^k) $F=\mathrm{GF}\left(p^k\right)$ be any finite field. Let G $G$ be a the group of all affine permutations on F $F$ (i.e. permutations of form ax+b)$x\mapsto ax+b$). Then the set of all functions from F $F$ to \bar{F} $\bar{F}$ is a linear representation of G, $G$, where g(f)=f(gx). $g(f)(x)=f(gx)$.

What are all sub-representations of this representation? Is it possible to characterize them?

Note: that in this case gcd(|G|,F) $\mathrm{gcd}\left(\left|G\right|,F\right)$ not equal to 1.$1$.

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Sub-represintationsSub-representations of the affine group

Let F=GF(p^k) be any finite field. Let G be a group of all affine permutations on F (i.e. permutations of form ax+b). Then the set of all functions from F to \bar{F} is a linear represintation representation of G, where g(f)=f(gx).

What are all sub-represintations sub-representations of represintationthis representation? Does Is it possible to charitirize characterize them?

Note: that in this case gcd(|G|,F) not equal to 1.

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Sub-represintations of affine group

Let F=GF(p^k) be any finite field. Let G be a group of all affine permutations on F (i.e. permutations of form ax+b). Then the set of all functions from F to \bar{F} is linear represintation of G, where g(f)=f(gx).

What are all sub-represintations of represintation? Does it possible to charitirize them? Note: that in this case gcd(|G|,F) not equal to 1.