Transitive permutation groups and equipotent voting rules

Let F a two-alternative voting rule and G_F the subgroup of permutations of the voters under which F is invariant. Group theoretic properties of G_F provide information about the voting rule F. In particular, sets of imprimitivity of G_F describe the ‘committee decomposition’ structure of F and permutation group transitivity of G_F (equipotency) is shown to be closely connected with equal distribution of power among the voters. If equipotency replaces anonymity in the hypotheses of May's theorem, voting rules other than simple majority are possible. By combining equipotency with two additional social choice conditions a new characterization of simple majority rule is obtained. Equipotency is proposed as an important alternative to the more restrictive anonymity as a fairness criterion in social choice.     

Transitive permutation groups in which all derangements are involutions

Let G be a transitive permutation group in which all derangements are involutions. We prove that G is either an elementary abelian 2-group or is a Frobenius group having an elementary abelian 2-group as kernel. We also consider the analogous problem for abstract groups, and we classify groups G with a proper subgroup H such that every element of G not conjugate to an element of H is an involution.     

An inequality for finite permutation groups

An inequality which has a combinatorial theoretical flavor is proved for finite permutation groups.     

Abelian Carter subgroups in finite permutation groups

We show that a finite permutation group containing a regular abelian self-normalizing subgroup is soluble.     

A measuring argument for finite permutation groups

LetG be a finite transitive permutation group on a finite setS. LetA be a nonempty subset ofS and denote the pointwise stabilizer ofA inG byC _G (A). Our main result is the following inequality: [G :C _G (A)]≥|G|^|A|/|S|. This paper is a part of the author’s Ph.D. thesis research, carried out at Tel Aviv University under the supervision of Professor Marcel Herzog.