Let G be an uncountable universal locally finite group. We study subgroups H < G such that for every g ∈ G, |H : H ∩H| < |H|.

DEFINITION. A group G is called a universal locally finite central extension of A provided that the following conditions are satisfied. (i) A <= (G (the centre of G). (ii) G is locally finite. (iii) (/1-injectivity). Suppose that A <= B <= D with A a (D, that D/A is finite, and that q>: B -> G is an ^-isomorphism (that is, q>{a) = a for all as A). Then there exists an extension q>: D -*• G of (...

Every group is an outer automorphism group of a locally finite p-group. This extends an earlier result [3] about countable outer automorphism groups. It is also in sharp contrast to results concerning the existence of outer automorphisms of nilpotent groups in [6, 13, 14].

We prove that there exist k in and 0 < epsilon in such that every non-abelian finite simple group G, which is not a Suzuki group, has a set of k generators for which the Cayley graph Cay(G; S) is an epsilon-expander.

The Algebraicity Conjecture treats model-theoretic foundations of algebraic group theory. It states that any simple group of finite Morley rank is an algebraic group over an algebraically closed field. In the mid-1990s a view was consolidated that this project falls into four cases of different flavour: even type, mixed type, odd type, and degenerate type. This book contains a proof of the conj...

Analogous to the way the integers can be decomposed uniquely into prime factors, finite groups can be decomposed into finite simple groups, which cannot be further decomposed nontrivially. We will prove the JordanHölder Theorem and show that finite groups can always be decomposed uniquely into finite simple groups, and discuss the classification of these groups. In particular, we will discuss t...