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The following theorem (which does not take the order $N$ of the group $G$ into account) shows that all possible combinations of $a$, $b$ and the order of $xy$ are possible. See Theorem 1.64 from Milne …

I'm not sure whether my answer is conceptual in your sense, but here is a relatively short proof. First of all, your definition of $f$ suggests the notation $$s_p := \sum_{i=p}^n x_i.$$ Now consider t …

The key word here is "polarity". A polarity of a projective plane with point set $P$ and line set $L$ is a map $\pi$ from $P \cup L$ to itself mapping points to lines and lines to points, such that $\ …

If you're only interested in finite permutation groups, then Koen S has given you the answer you needed. If you allow infinite objects, then there are much more symmetric objects than S(5,8,24). In f …

I might be missing something, but it seems to me that there is not much going on in your construction. In fact, your original action of $Z_n$ on $X$ does nothing more than putting a cyclic ordering on …

(Reposting my comment as an answer, as requested by the OP.) If you have a solution with $a_i \in \{−1,1\}$, then you also have a solution with $a_i \in \{0,1\}$ simply by replacing each $a_i$ with $ …