### 2-transitiv abstract ovals of odd order.

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In the class of complete games, the Shapley index of power is the characteristic invariant of the group of automorphisms, for these are exactly the permutations of players preserving the index.

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, tetravalent one-regular graphs of order 3p², where p is a prime, are classified.

A solvable primitive group with finitely generated abelian stabilizers is finite.

A graph $X$, with a group $G$ of automorphisms of $X$, is said to be $(G,s)$-transitive, for some $s\ge 1$, if $G$ is transitive on $s$-arcs but not on $(s+1)$-arcs. Let $X$ be a connected $(G,s)$-transitive graph of prime valency $p\ge 5$, and ${G}_{v}$ the vertex stabilizer of a vertex $v\in V\left(X\right)$. Suppose that ${G}_{v}$ is solvable. Weiss (1974) proved that $|{G}_{v}{|\mid p(p-1)}^{2}$. In this paper, we prove that ${G}_{v}\cong ({\mathbb{Z}}_{p}\u22ca{\mathbb{Z}}_{m})\times {\mathbb{Z}}_{n}$ for some positive integers $m$ and $n$ such that $ndivm$ and $m\mid p-1$.