We construct a family of infinite, non-locally finite highly arc-transitive digraphs which do not have universal reachability relation and which omit special digraphs called ‘crowns’. Moreover, there is no homomorphism from any of our digraphs onto Z. The methods are adapted from [5] and [6].

A graph is said to be symmetric if its automorphism group is transitive on its arcs. A complete classification is given of pentavalent symmetric graphs of order 24p for each prime p. It is shown that a connected pentavalent symmetric graph of order 24p exists if and only if p=2, 3, 5, 11 or 17, and up to isomorphism, there are only eleven such graphs.

We started our research with the intent on answering the following question: can we find a way to calculate all the subgroups of the symmetric group. This is easier said that done, as the number of subgroups for a symmetric group grows quickly with each successive symmetric group. This problem can actually be simplified to finding the subgroup conjugacy classes. So now we have the slightly diff...

Recall that, given a set X, the set SX of all bijections from X to itself (or, more briefly, permutations of X) is group under function composition. In particular, for each n ∈ N, the symmetric group Sn is the group of permutations of the set {1, . . . , n}, with the group operation equal to function composition. Thus Sn is a group with n! elements, and it is not abelian if n ≥ 3. If X is a fin...

the conditions under which, multilinear forms (the symmetric case and the non symmetric case),can be written as a product of linear forms, are considered. also we generalize a result due to s.kurepa for 2n-functionals in a group g.