On asymptotic dimension of groups.
The group of all automorphisms leaving invariant every subnormal subgroup of the group is studied. In particular it is proved that is metabelian if is soluble, and that is either finite or abelian if is polycyclic.
Using a lemma on subnormal subgroups, the problem of nilpotency of multiplication groups and inner permutation groups of centrally nilpotent loops is discussed.
Let be a maximal -order in a division quaternion algebra over which is split at the place . The present article gives an algorithm to compute a fundamental domain for the action of the group of units on the Bruhat-Tits tree associated to . This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group in terms of generators and relations. Moreover we determine an upper bound...
An algebraic structure is said to be congruence permutable if its arbitrary congruences and satisfy the equation , where denotes the usual composition of binary relations. To an arbitrary -set satisfying , we assign a semigroup on the base set containing a zero element , and examine the connection between the congruence permutability of the -set and the semigroup .
In this paper, we have studied the connectedness of the graphs on the direct product of the Weyl groups. We have shown that the number of the connected components of the graph on the direct product of the Weyl groups is equal to the product of the numbers of the connected components of the graphs on the factors of the direct product. In particular, we show that the graph on the direct product of the Weyl groups is connected iff the graph on each factor of the direct product is connected.