The algebraicity conjecture for simple groups of finite Morley rank, also known as the Cherlin-Zilber conjecture, states that simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. In the last 15 years, the main line of attack on this problem has been Borovik’s program of transferring methods from finite group theory. Borovik’s program has led to consi...

In this paper, we study the probability that a randomly chosen element in a finite group has a square root, in particular the simple groups of Lie type of rank 1, the sporadic finite simple groups and the alternating groups.

A group is called simple if it has more than one element and if its only normal subgroups are the trivial subgroup and the whole group. In other words, it is a group with precisely two normal subgroups. It is an elementary exercise to show that the abelian simple groups are precisely the finite cyclic groups of prime order. The nonabelian simple groups, on the other hand, are far more complicat...

The spectrum ?(G) of a finite group G is the set orders its elements. following sufficient criterion nonsolvability proved: if, among prime divisors order G, there are four different primes such that contains all their pairwise products but not product any three these numbers, then nonsolvable. Using this result, we show for q ? 8 and ? 32, direct square Sz(q) × simple exceptional Suzuki unique...

In a connected group of finite Morley rank in which the generic element belongs to a connected nilpotent subgroup, proper normalizing cosets of definable subgroups are not generous. We explain why this is true and what consequences this has on an abstract theory of Weyl groups in groups of finite Morley rank. The only known infinite simple groups of finite Morley rank are the simple algebraic g...