a positive integer $n$ is called a clt number if every group of order $n$ satisfies the converse of lagrange's theorem‎. ‎in this note‎, ‎we find all clt and supersolvable numbers up to $1000$‎. ‎we also formulate some questions about the distribution of these numbers‎.

By the Artin Induction theorem,C(G) is a finite abelian group with exponent dividing the order of G. Some work on this sequence has already been done. In [14] and [16], Ritter and Segal proved that C(G) = 0 for G a finite p–group. Serre [17, p. 104] remarked that C(G) / = 0 for G = Z/3 × Q8 (the direct product of a cyclic group of order 3 and a quaternion group of order 8). Berz [2] gave a nice...

For G a finite group and p a prime this paper proves two theorems under hypotheses that restrict the index of the subgroup generated by every p-element x in certain subgroups generated by pairs of its conjugates. Under one set of hypotheses G is shown to be supersolvable. Simple groups satisfying a complementary fusion-theoretic hypothesis are classified.

We show that the subgroup lattice of any finite group satisfies Frankl’s UnionClosed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable and dually semimodular lattices. A common technical result used to prove both may be of some independent interest.

We study a class of hyperplane arrangements associated to complementary signed graphs in which the positive part and the negative part are complementary to each other in Kn, the complete graph on n vertices. These arrangements form a subclass of the Dn arrangement but do not contain the An−1 arrangement. The main result says that the arrangement A(G) of a complementary signed graph G is superso...

It is a well known fact that a supersolvable lattice is ELoshellable. Hence a supersolvable lattice (resp., its Stanley-Reisner ring) is Cohen-Macaulay. We prove that if L is a supersolvable lattice such that all intervals have non-vanishing Mt~bius number, then for an arbitrary element x e L the poser L {x} is also Cohen-Macaulay. Posets with this property are called 2-Cohen-Macaulay posets. I...

One way to view Theorem 1.1 is as a statement that the algebraic structure of a finitely generated profinite group somehow also encodes the topological structure. That is, if one wishes to know the open subgroups of a profinite group G, a topological property, one must only consider the subgroups of G of finite index, an algebraic property. As profinite groups are compact topological spaces, an...