The class of finitely presented algebras over a field K with a set of generators a1, ..., an and defined by homogeneous relations of the form a1 a2 ⋯ an = aσ (1) aσ (2) ⋯ aσ (n), where σ runs through a subset H of the symmetric group Symn of degree n, is introduced. The emphasis is on the case of a cyclic subgroup H of Symn of order n. A normal form of elements of the algebra is obtained. It is shown that the underlying monoid, defined by the same (monoid) presentation, has a group of fractions and this group is described. Properties of the algebra are derived. In particular, it follows that the algebra is a semiprimitive domain. Problems concerning the groups and algebras defined by arbitrary subgroups H of Symn are proposed. © 2009 Elsevier B.V. All rights reserved.