For a ZG-lattice A, the nth partial free Euler characteristic εn(A) is defined as the infimum of all where F* varies over all free resolutions of A. It is shown that there exists a stably free resolution E* of A which realises εn(A) for all n≥0 and that the function n → εn(A) is ultimately polynomial no residue classes. The existence of E* is established with the help of new invariants σn(A) of A. These are elements in certain image groups of the projective class group of ZG. When ZG allows cancellation, E* is a minimal free resolution and is essentially unique. When A is periodic, E* is ultimately periodic of period a multiple of the projective period of A.

The following question is discussed and evidence for and against it is advanced: is it true that if F is an arbitrary finite subgroup of an arbitrary non-linear simple locally finite group G, then CG(F) is infinite? The following points to an affirmative answer.

Theorem A. Let F be an arbitrary finite subgroup of a non-linear simple locally finite group G. Then there exist subgroups D ◃ C ≤ G such that F centralizes C/D, F∩C ≤ D, and C/D is a direct product of finite alternating groups of unbounded orders. In particular, F centralizes an infinite section of G.

Theorem A is deduced from a “local” version, namely

Theorem B. There exists an integer valued function f(n, r) with the following properties. Let H be a finite group of order at most n, and suppose that H ≤ S, where S is either an alternating group of degree at least f = f(n, r) or a finite simple classical group whose natural projective representation has degree at least f. Then there exist subgroups D ◃ C ≤ S such that (i) [H, C] ≤ D, (ii) H ∩ C ≤ D, (iii) C/D ≅ Alt(r), (iv) D = 1 if S is alternating, and D is a p-group of class at most 2 and exponent dividing p2 if S is a classical group over a field of characteristic p.

The natural “local version” of our main question is however definitely false.

Proposition C. Let p be a given prime. Then there exists a finite group H that can be embedded in infinitely many groups PSL(n, p) as a subgroup with trivial centralizer.

The known characterization of the Mathieu group M12 by the structure of the centralizer of a 2-central involution is based on the application of the theory of exceptional characters and uses in addition a block theoretic result which asserts that a simple group of order |M12| is isomorphic to M12. The details of the proof of the latter result had never been published. We show here that M12 can be handled in a completely elementary and group theoretical way.

We discuss some general properties and limitations of the concept of outer Fitting pairs introduced earlier by the author. We describe an outer Fitting pair as a co-cone in the category of what we call outer groups (roughly speaking the category of groups modulo inner automorphisms). It is shown that generally no universal outer Fitting pair exists, whence this category is not co-complete. Additionally it is shown that if the target group of an outer Fitting pair is finite, then the much more amenable concept of normal Fitting pairs (that is, co-cones in the category of groups) applies.

Let G be a finite group of even order coprime to 3. If G admits a fixed-point-free automorphism group isomorphic to the symmetric group on three letters, then we prove that G is soluble.

Two subgroups ME(G) and MI(G) of the Schur multiplier M(G) of a finite group G are introduced: ME(G) contains those cohomology classes [α] of M(G) for which every element of G is α-regular, and MI(G) consists of those cohomology classes of M(G) which contain a G-invariant cocycle. It is then shown that under suitable circumstances, such as when G has odd order, that each element of MI(G) can be expressed as the product of an element of ME(G) and an element of the image of the inflation homomorphism from M(G/G′) into M(G).

Finite ρ-groups with all of their maximal subgroups isomorphic are studied by means of the coclass. All such groups of coclass I and 2 are determined, while those of coclass 3 are shown to have order at most ρ13. A general bound for the order is given as a function of ρ and the coclass only.

We give presentations for the groups PSL(2, pn), p prime, which show that the deficiency of these groups is bounded below. In particular, for p = 2 where SL(2, 2n) = PSL(2, 2n), we show that these groups have deficiency greater than or equal to – 2. We give deficiency – 1 presentations for direct products of SL(2, 2n) for coprime ni. Certain new efficient presentations are given for certain cases of the groups considered.

We study the characteristic p analogue of M-groups, the so-called Mp-group Generalizing this notion, we also consider the condition that the modular irreducible representations are induced from representations of dimension < p, or even weaker, of dimension not divisible by p.

All subnormal subgroups of hypernormalizing groups have by definition subnormal normalizers. It is shown that finite soluble HN-groups belong to the class of groups of Fitting length three. Finite HN-groups are considered including those with subnormal quotient isomorphic to SL(2,5).

In this note we present a general Jordan-Hölder type theorem for modular lattices and apply it to obtain various (old and new) versions of the Jordan-Hölder Theorem for finite groups.

In this note we introduce a self-centralizing characteristic subgroup, associated with quasinilpotent injectors, of a finite group.

Let G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.

Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

In this paper we show that a group A is embedded in any finite group G as a subnormal subgroup with low degree of complication, provided that the automorphism group of A satisfies a condition depending on some Fitting class (which coincides with completeness for the Fitting class of all groups). A criterion is given for these groups as to whether they can be embedded subnormally in the commutator subgroup of some finite group or not.

We give new presentations of the five Mathieu groups, the simple groups J1, J2, HS, McL, Co3, and some other simple and related groups. All generators in these presentations are involutions. Our presentations are simpler than the known presentations of this type for the groups mentioned above.

We study the embeddings of a finite p-group U into Sylow p-subgroups of Sym (U) induced by the right regular representation p: U→ Sym(U). It turns out that there is a one-to-one correspondence between the chief series in U and the Sylow p-subgroups of Sym (U) containing Up. Here, the Sylow p-subgroup Pσ of Sym (U) correspoding to the chief series σ in U is characterized by the property that the intersections of Up with the terms of any chief series in Pσ form σp. Moreover, we see that p: U→ Pσ are precisely the kinds of embeddings used in a previous paper to construct the non-trivial countable algebraically closed locally finite p-groups as direct limits of finite p-groups.

We determine all conjugacy classes of maximal local subgroups of Thompson's sporadic simple group, and all maximal non-local subgroups except those with socle isomorphic to one of five particular small simple groups.

We determine all conjugacy classes of maximal p-local subgroups of the Monster for p ≠ 2.

It is proved that a finite soluble group of order n has at most (n − 1)/(q − 1) maximal subgroups, where q is the smallest prime divisor of n.

A group G is called normally (subnormally) detectable if the only normal (subnormal) subgroups in any direct product G1 × … × Gn of copies of G are just the direct factors Gi. We give an internal characterization of finite subnormally detectable groups and obtain analogous results for associative rings and for Lie algebras. The main part of the paper deals with a study of normally detectable groups, where we verify a conjecture of T. O. Hawkes in a number of special cases.