We examine the projective dimensions of Mackey functors and cohomological Mackey functors. We show over a field of characteristic p that cohomological Mackey functors are Gorenstein if and only if Sylow p-subgroups are cyclic or dihedral, and they have finite global dimension if and only if the group order is invertible or Sylow subgroups are cyclic of order 2. By contrast, we show that the onl...

A subgroup H is said to be s-permutable in a group G, if HP = PH holds for every Sylow subgroup P of G. If there exists a subgroup B of G such that HB = G and H permutes with every Sylow subgroup of B, then H is said to be SS-quasinormal in G. In this paper, we say that H is a weakly SS-quasinormal subgroup of G, if there is a normal subgroup T of G such that HT is s-permutable and H ∩ T is SS-...

Let G be a subgroup of S,, given in terms of a generating set of permutations, and let p be a prime divisor of 1 G 1. If G is solvable-and, more generally, if the nonabelian composition factors of G are suitably restricted-it is shown that the following can be found in polynomial time: a Sylow p-subgroup of G containing a given p-subgroup, and an element of G conjugating a given Sylow p-subgrou...

Even Type Conjecture. Let G be a simple group of finite Morley rank of even type. Then G is a Chevalley group over an algebraically closed field of characteristic two. See [16] for an informal introduction to the subject, [1] for a recent survey of the classification programme, and [17] for general background on groups of finite Morley rank. An infinite simple group G of finite Morley rank whos...