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### The degree of a constant polynomial on a finite group

If $X = A_5$ then $\exp(X) = \mathop{\rm lcm}(2,3,5) = 30$ but $\deg 1_X \leq 10$. Indeed if $c \in X$ is a 5-cycle then $$f(x) := x c x c^2 x c^3 x c^4 x$$ satisfies $f(x)^2 = 1$ for all $x \in X$ ...
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