Upper and lower estimates for the maximal commutator length of a noncentral element elementary subgroup general linear group over skew-field are proved on basis an multiplicative this skew-field.

We offer a sufficient condition for a closed subset in an association scheme to be maximal. The result generalizes naturally the well-known (group-theoretical) fact that the one-point-stabilizer of a flag transitive automorphism group of a 2-design with *=1 must be a maximal subgroup. In finite group theory, there exist a lot of (important) sufficient conditions for a subgroup to be maximal. In...

Let K be a global field and let OK,S be the ring of S-integers of K for some finite set S of primes of K. We prove that whatever the infinite subset E ⊆ OK,S and the polynomial f(X) ∈ K[X], the subsets E and f(E) have the same number of residual classes modulo m for almost all maximal ideals m of OK,S if and only if deg(f) = 1 when the characteristic of K is 0 and f(X) = g(Xp k ) for some integ...

We discuss the following conjecture of Kitaoka: if a finite subgroup G of GLn(OK) is invariant under the action of Gal(K/Q) then it is contained in GLn(K ). Here OK is the ring of integers in a finite, Galois extension K of Q and K ab is the maximal, abelian subextension of K. Our main result reduces this conjecture to a special case of elementary abelian p−groups G. Also, we construct some new...

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...

The smallest known thick generalized octagon has order (2,4) and can be constructed from the parabolic subgroups of the Ree group 2F4(2). It is not known whether this generalized octagon is unique up to isomorphism. We show that it is unique up to isomorphism among those having a point a whose stabilizer in the automorphism group both fixes setwise every line on a and contains a subgroup that i...