In this paper we investigate the minimum number of maximal subgroups H i , i = 1, . . . ,k of the symmetric group S n (or the alternating group A n ) such that each element in the group S n (respectively A n ) lies in some conjugate of one of the H i . We prove that this number lies between aφ(n) and bn for certain constants a, b, where φ(n) is the Euler phi-function, and we show that the number depends on the arithmetical complexity of n. Moreover in the case where n is divisible by at most two primes, we obtain an upper bound of 2 + φ(n)/2, and we determine the exact value for S n when n is odd and for A n when n is even.

In the standard arrovian framework and under the assumption that individual preferences and social outcomes are linear orders on the set of alternatives, we study the rules which satisfy suitable symmetries and obey the majority principle. In particular, supposing that individuals and alternatives are exogenously partitioned into subcommittees and subclasses, we provide necessary and sufficient conditions for the existence of reversal symmetric majority rules that are anonymous and neutral with respect to the considered partitions. We also determine a general method for constructing and counting those rules and we explicitly apply it to some simple cases.

In this paper we show that if n ≥ 5 and G is any of the groups SUn(q) with n = 6, Sp 2n (q) with q odd, Ω 2n+1 (q), Ω ± 2n (q), then G and the simple group G = G/Z(G) are not 2-coverable. Moreover the only 2-covering of Sp 2n (q), with q even, has components O − 2n (q) and O + 2n (q).

The normal covering number γ(G) of a finite, non-cyclic group G is the least number of proper subgroups such that each element of G lies in some conjugate of one of these subgroups. We prove that there is a positive constant c such that, for G a symmetric group Sym(n) or an alternating group Alt(n), γ(G) ≥ cn. This improves results of the first two authors who had earlier proved that aϕ(n) ≤ γ(G) ≤ 2n/3, for some positive constant a, where ϕ is the Euler totient function. Bounds are also obtained for the maximum size κ(G) of a set X of conjugacy classes of G = Sym(n) or Alt(n) such that any pair of elements from distinct classes in X generates G, namely cn ≤ κ(G) ≤ 2n/3.2000 Mathematics Subject Classification. 20B30.

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