A characterization of the Mathieu group $M_{12}$ is established; the characterization is used by Aschbacher and Smith in their classification of the quasithin finite simple groups.

A group is called capable if it is a central factor group. Let ${\mathcal{P}}$ denote the class of finite $p$-groups of odd order and nilpotency class 2. In this paper we determine the capable 2-generator groups in ${\mathcal{P}}$. Using the explicit knowledge of the nonabelian tensor square of 2-generator groups in ${\mathcal {P}}$, we first determine the epicenter of these groups and then identify those with trivial epicenter, making use of the fact that a group has trivial epicenter if and only if it is capable. A capable group in ${\mathcal{P}}$ has the two generators of highest order in a minimal generating set of equal order. However, this condition is not sufficient for capability in ${\mathcal{P}}$. Furthermore, various homological functors, among them the exterior square, the symmetric square and the Schur multiplier, are determined for the 2-generator groups in ${\mathcal{P}}$.

We consider certain properties of finite groups in which the subnormal subgroups permute with all the Sylow subgroups. Such groups are called PST-groups. If $G$ is such a group and $ H_1 / K_1 $ and $ H_2 / K_2 $ are isomorphic abelian chief factors of $G$ such that $ H_1 H_2 \subseteq G' $, then they are operator isomorphic. Moreover, if all the abelian isomorphic chief factors of a PST-group $G$ are operator isomorphic, then all the subnormal subgroups are hypercentrally embedded in $G$.

We establish many new properties of $M''$-groups and give a large set of new counter-examples to the well-known problems of R. Baer (1949) and S. N. Chernikov (1959) concerning socle groups, $M''$-groups and $M'$-groups. In passing we also show that for a periodic $FC$-group $G$ and its locally soluble radical $R$ the factor group $G/L(G)R$ is residually finite, where $L(G)$ is the product of all normal semisimple subgroups of $G$.

It is well known that a product $ab$ in a ring may have an inverse without $ba$ being invertible. However, if $ab$ has a quasi-inverse, then so does $ba$. This note provides a (3-line) proof and an explanation.

In earlier work, the authors described the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group. As might be expected, the even order case is considerably more complicated and we have found it necessary to divide it into two parts. This part deals with the situation where we have a finite group $G$ with a cyclic permutable subgroup $A$ satisfying the additional hypothesis that $X$ is permutable in $A_2X$ for all cyclic subgroups $X$ of $G$ (where $A_2$ is the $2$-component of $A$).

In the description of the group of lattice automorphisms of modular groups, certain locally finite modular $p$-groups of finite exponent play a basic role. In the present paper significant structural properties of the group of autoprojectivities of such groups are investigated and placed in evidence.

There exists a unique symmetric (36,15,6) design $\mathcal{D}$ having $G'(2,2) \cong U(3,3)$ as an automorphism group. There is an incidence matrix $M$ of $\mathcal{D}$ which is symmetric with $1$ everywhere on the main diagonal. Thus $\mathcal{D}$ admits a polarity for which all points are absolute. Therefore, $M$ is an adjacency matrix of a strongly regular graph with parameters $(36,14,4,6)$.

Using this design one can produce a series of symmetric designs with parameters $(4 \cdot (3 \cdot 2^k)^2,2 \cdot (3 \cdot 2^k)^2 - 3 \cdot 2^k, (3 \cdot 2^k)^2 - 3 \cdot 2^k)$, $k \in N$, each of which admits an automorphism group isomorphic to the unitary group $U(3,3)$. There is an incidence matrix for each of these designs which is symmetric with constant diagonal. Therefore, these matrices correspond to adjacency matrices of strongly regular graphs.

We classify the (finite) $p$-groups of maximal class that are isomorphic to the full automorphism group of a (finite or infinite) group. The only such $p$-groups are the nonabelian groups of order $8$ and 3-groups in a certain family, whose structure is fully described. Up to isomorphism there is exactly one such 3-group for each even nilpotency class greater than $2$, and none for other classes.

A group $G$ satisfies the maximal condition on non FC-subgroups if every ascending chain of non FC-subgroups terminates in finitely many steps. In this paper the authors obtain some structural results for locally FC-groups with the maximal condition on non FC-subgroups.

It is proved consistent with ZFC + GCH that for every Whitehead group $A$ of infinite rank, there is a Whitehead group $H_{A}$ such that $\operatorname{Ext} (H_{A},A)\neq 0$. This is a strong generalization of the consistency of the existence of non-free Whitehead groups. A consequence is that it is undecidable in ZFC + GCH whether every $\mathbb{Z}$-module has a $^{\perp }\{ \mathbb{Z\}}$-precover. Moreover, for a large class of $\mathbb{Z}$-modules $ N$, it is proved consistent that a known sufficient condition for the existence of $^{\perp }\{N\}$-precovers is not satisfied.

A theorem of B.H. Neumann states that each subgroup of a group $G$ has finite index in a normal subgroup of $G$ if and only if the commutator subgroup $G'$ of $G$ is finite, i.e., $G$ is finite-by-abelian. As a group lattice version of this theorem for a periodic group $G$, it is proved that each subgroup of $G$ has finite index in a modular subgroup of $G$ if and only if $G$ is an extension of a finite group by a group with modular subgroup lattice.

Reinhold Baer's contributions to and influence on the theory of abelian groups are surveyed, concentrating on his publications on the theory of $p$-groups, torsion-free and mixed groups, subgroup lattices, endomorphism rings, the group Ext, as well as injective modules. His papers on these subjects were instrumental in the development of this branch of group theory.

In this paper we are mainly concerned with the class $\mathcal{LN}$ of all locally nilpotent groups. Using similar arguments as in [grs] we first show that there is no universal group in $\mathcal{LN}_\lambda$ if $\lambda$ is a cardinal such that $\lambda=\lambda^{\aleph_0}$; here we call a group $G$ universal (in $\mathcal{LN}_\lambda$) if any group $H\in\mathcal{LN}_\lambda$ can be embedded into $G$, where $\mathcal{LN}_\lambda$ denotes the class of all locally nilpotent groups of cardinality at most $\lambda$. However, our main interest is in the construction of torsion-free epi-universal groups in $\mathcal{LN}_\lambda$, where $G\in\mathcal{LN}_\lambda$ is said to be epi-universal if any group $H\in\mathcal{LN}_\lambda$ is an epimorphic image of $G$. Thus we give an affirmative answer to a question of Plotkin. To prove the torsion-freeness of the constructed locally nilpotent group we adjust the well-known commutator collecting process due to P. Hall to our situation. Finally, we briefly discuss how to apply the methods we used for the class $\mathcal{LN}$ to other canonical classes of groups to construct epi-universal objects.

The Baer splitting problem from the 1930s is revisited, after which, using current knowledge about maximal Cohen-Macaulay modules, the structure of Baer modules over regular integral domains of higher Krull dimension is explored. In particular, the countably generated ones in the local case are shown to be free.

It is shown that finite groups with Schur multipliers of large exponent lead to groups of central type.

We survey some recent results related to the finite basis question for varieties of groups.

We revisit Reinhold Baer's work on equivalent extensions, which can be considered as a forerunner of the authors' series of equivalence theorems. Our focus is on a paper entitled Extension Types of Abelian Groups published by Baer in 1949. In this paper, the main results were for a rather restrictive class of extensions called little extensions, but the notion of two extensions of A by B being equivalent given there are generally applicable. Our theme here is that Baer's vision and understanding of extensions placed him much ahead of the time in which he studied the subject in the 1930's and '40's.

We prove that every noncyclic subgroup of a free $m$-generator Burnside group $B(m,n)$ of odd exponent $n \gg 1$ contains a subgroup $H$ isomorphic to a free Burnside group $B(\infty,n)$ of exponent $n$ and countably infinite rank such that, for every normal subgroup $K$ of $H$, the normal closure $\langle K \rangle^{B(m,n)}$ of $K$ in $B(m,n)$ meets $H$ in $K$. This implies that every noncyclic subgroup of $B(m,n)$ is $\operatorname{SQ}$-universal in the class of groups of exponent $n$.

The subgroups studied in this paper are generalizations of the subgroup $R_2(G) =\{ x\in G |[x,g,g]=1, \forall g\in G\}$ of right 2-Engel elements of $G$. It is shown that they are actually partial margins and their embedding in $G$ is investigated.

A modification is suggested for a 1934 example of Reinhold Baer which shows that not all homomorphisms $C\to(\operatorname{Aut} A)/(\operatorname{Inn}\space A)$ arise from conjugation action in group extensions $1\to A\to B\to C\to 1.$

For certain artinian modules over group rings, we obtain the Baer decomposition, that is, a direct decomposition into two summands such that all chief factors of the first summand are ${\mathcal X}$-central and all chief factors of the second summand are ${\mathcal X}$-eccentric, for some formation $\mathcal X$ of finite groups.

A non-finitary group $G$ is said to be an $\mathrm{\small L}\mathfrak{A}$-group if it is a direct limit of finite alternating groups $G_i=\operatorname{Alt}({\Omega_i})$ $(i\!\in\!I)$ such that each $G_i$ has only trivial or natural orbits on the sets $\Omega_j$ $(j\! \gt \!i)$. We determine the confined subgroups of $\mathrm{\small L}\mathfrak{A}$-groups and relate them naturally to the ideals in the group algebra $\mathbb{K} G$ over any field $\mathbb{K}$ of characteristic zero. Moreover, we show that the non-trivial ideals in $\mathbb{C} G$ can be related to normalized positive definite class functions $f\colon G\to\mathbb{C}$ if and only if the number of $G_i$-orbits in $\Omega_j$ $(j\! \gt \!i)$ is asymptotically a linear function of $|\Omega_j|$ for all $i$.

We elaborate on a method of G. Michler [Mi] to construct finite groups with a prescribed involution centralizer $H$ using compatible pairs and amalgamation of $H$ and another naturally arising $2$-local subgroup. Here we deal with the particular set-up leading to the simple groups $GL(5,2), M_{24}$ and $He$.

In this paper we determine all finite simple groups $G$ for which the automorphism group $\operatorname{Aut} G$ splits over $G=\operatorname{Inn} G$.

Let $H$ be a finite group having center $Z(H)$ of even order. By the classical Brauer-Fowler theorem there can be only finitely many non isomorphic simple groups $G$ which contain a $2$-central involution $t$ for which $C_G(t) \cong H$.

In this article we give several conditions which together suffice to prove that up to isomorphism there is a unique simple group $G$ having a $2$-central involution $z$ with centralizer $C_G(z) \cong H$. Together they yield a practical uniqueness criterion (Theorem 2.1). This is demonstrated by giving a new short uniqueness proof for Janko's sporadic simple group $J_1$. Similar, uniform uniqueness proofs are outlined for ten other sporadic simple groups.

A permutation group is a three-star group if it induces a non-trivial group on each $3$-element subset of points. Our main results are that a primitive three-star group is generously transitive and that a finite primitive three-star group has rank at most $3$, that is, a stabiliser has at most $3$ orbits. We also describe the structure of an arbitrary (non-primitive) three-star group and give a collection of examples. In particular, we sketch a construction of infinite primitive three-star groups of arbitrarily high rank.

We prove a structure theorem for a class of finite transitive permutation groups that arises in the study of finite bipartite vertex-transitive graphs. The class consists of all finite transitive permutation groups such that each non-trivial normal subgroup has at most two orbits, and at least one such subgroup is intransitive. The theorem is analogous to the O'Nan-Scott Theorem for finite primitive permutation groups, and this in turn is a refinement of the Baer Structure Theorem for finite primitive groups. An application is given for arc-transitive graphs.

If $G$ is a group with an injective endomorphism $\phi$, then the HNN-extension $G_\phi = \langle G, t \,:\, t^{-1}gt=g \phi \,\, \mbox{for all}\,\, g \in G\rangle $ is called the ascending HNN-extension of $G$ determined by $\phi$. We prove that $G_\phi$ is residually finite when $G$ is either finitely generated abelian-by-polycyclic-by-finite or reduced soluble-by-finite minimax. We also provide an example of a $3$-generator residually finite soluble group $G$ of derived length $3$ with a non-residually-finite ascending HNN-extension.

The real plane is a Baer subplane of the complex projective plane in the following sense: each complex line contains a real point and, dually, each complex point is on a real line. In the first part of this survey main aspects of the general theory will be reviewed. The second part is concerned with finite planes. Part 3 deals rather completely with Baer subplanes of compact, connected topological projective planes.

Let $L$ be a lattice. A lattice is called $L$-free if it has no sublattice isomorphic to $L$. In this paper we study finite groups whose subgroup lattices are $L$-free for certain lattices $L$.

Enochs' proof of the Flat Cover Conjecture [BEE] is based on a construction of special preenvelopes from [ET1]. A recent result of Eklof and Shelah [ES] implies consistency (with ZFC + GCH) of non-existence of the dual construction of special precovers, for certain abelian groups. By an analysis of Ext on limits of well-ordered inverse systems, we prove that a weaker form of the dual construction is still available (in ZFC), for any module over any ring.

The main result of the paper states that a metabelian group $G$ has deviation for normal subgroups if and only if it has a finite series of normal subgroups each of whose factor meets the maximal or the minimal condition for $G$-invariant subgroups.

Let $M$ be a module over the commutative ring $R$. We consider the group $G$ of all automorphisms $g$ of $M$ for which $M(g-1)$ is $R$-Artinian. We show that $G$ has a locally residually nilpotent normal subgroup modulo which $G$ is a subdirect product of finitary linear groups over field images of $R$. This can be used to study certain subgroups of $G$. For example, if $H$ is a locally finite subgroup of $G$, then $H$ is isomorphic to a finitary linear group of characteristic zero if $R$ is an algebra over the rationals and $H/O_p(H)$ is isomorphic to a finitary linear group of characteristic the prime $p$ if R has characteristic a power of $p$. It also gives information about $\operatorname{Aut}_RM$ if $M$ itself is $R$-Artinian.