Let K[G] be the group algebra of a locally finite group G over a field K of characteristic p > 0. In this paper, we show that K[G] is semiprimitive if and only if G has no locally subnormal subgroup of order divisible by p. Thus we settle the semiprimitivity problem for such group algebras by verifying a conjecture which dates back to the middle 1970’s. Of course, if G has a locally subnormal s...

The author considers the problem of determining whether certain classes of lattice ordered groups (¿-groups) have the amalgamation property. It is shown that the classes of abelian totally ordered groups (ogroups) and abelian /-groups have the property, but that the class of ¿-groups does not. However, under certain cardinality restrictions one can find an ¿-group which is the "product" of /-gr...

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. One of the major theorems in the area is Borovik’s trichotomy theorem. The ‘trichotomy’ here is a case division of the generic minimal counterexamples within odd type, that is, groups with a large and divisible Sylow◦ 2-subgroup. The so-called ‘u...

The following basic results on infinite locally finite subgroups of a free m-generŽ . 48 ator Burnside group B m, n of even exponent n, where m ) 1 and n G 2 , n is divisible by 29, are obtained: A clear complete description of all infinite groups that Ž . Ž . are embeddable in B m, n as maximal locally finite subgroups is given. Any Ž . infinite locally finite subgroup L of B m, n is contained...

We calculate the abelianizations of the level L subgroup of the genus g mapping class group and the level L congruence subgroup of the 2g × 2g symplectic group for L odd and g ≥ 3. Historical note. I originally wrote this paper in March of 2008. Towards the end of that month, I gave a master class on the Torelli group at the University of Aarhus. That master class ended in a conference, and I h...

Let $G$ be a finite group. A subgroup $H$ is called $S$-semipermutable in if $HG_p$ = $G_pH$ for any $G_p\in Syl_p(G)$ with $(|H|, p) 1$, where $p$ prime number divisible $|G|$. Furthermore, said to $NH$-embedded there exists normal $T$ of such that $HT$ Hall and $H \cap T \leq H_{\overline{s}G}$, $H_{\overline{s}G}$ the largest contained $H$, $SS$-quasinormal provided supplement $B$ permutes e...

It is known that if P is either the property w-bounded or countably compact, then for every cardinal a 2 w there is a P-group G such that H.G = a and no proper, dense subgroup of G is a P-group. What happens when P is the property pseudocompact? The first-listed author and Robertson have shown that every zero-dimensional Abelian P-group G with H.G > o has a proper, dense, P-group. Turning to th...

We investigate the implications of the separability principle for the class of problems allocating an infinitely divisible commodity among a group of agents with single-peaked preferences and no-participation option. The separability principle requires that for two problems with the same population, but possibly different social endowments, in which the preferences of agents may change, if ther...

By the Shepherd–Leedham-Green–McKay theorem on finite p-groups of maximal class, if a finite p-group of order pn has nilpotency class n−1, then it has a subgroup of nilpotency class at most 2 with index bounded in terms of p. Counterexamples to a rank analogue of this theorem are constructed, which give a negative solution to Problem 16.103 in Kourovka Notebook. Moreover, it is shown that there...