Let C G (X ) be the set of all (equivalence classes of) regular covering projections of a given connected graph X along which a given group G ≤ Aut X of automorphisms lifts. There is a natural lattice structure on C G (X ), where ℘ 1 ≤ ℘ 2 whenever ℘ 2 factors through ℘ 1 . The sublattice C G (℘) of coverings which are below a given covering ℘ :X → X naturally corresponds to a lattice N G (℘) of certain subgroups of the group of covering transformations. In order to study this correspondence, some general theorems regarding morphisms and decomposition of regular covering projections are proved. All theorems are stated and proved combinatorially in terms of voltage assignments, in order to facilitate computation in concrete applications.For a given prime p, let C p G (X ) ≤ C G (X ) denote the sublattice of all regular covering projections with an elementary abelian p-group of covering transformations. There is an algorithm which explicitly constructs C p G (X ) in the sense that, for each member of C p G (X ), a concrete voltage assignment on X which determines this covering up to equivalence, is generated. The algorithm uses the well known algebraic tools for finding invariant subspaces of a given linear representation of a group. To illustrate the method two nontrival examples are included.

An enumeration result for orientably regular hypermaps of a given type with automorphism groups isomorphic to PSL(2, q) or PGL(2, q) can be extracted from a 1969 paper by Sah. We extend the investigation to orientable reflexible hypermaps and to nonorientable regular hypermaps, providing many more details about the associated computations and explicit generating sets for the associated groups.2000 Mathematics subject classification: primary 57M15; secondary 05C25, 20F05. Keywords and phrases: hypermap, regular hypermap, triangle group, projective linear group. IntroductionA regular hypermap H is a pair (r, s) of permutations generating a regular permutation group on a finite set, and provides a generalization of the geometric notion of a regular map on a surface, by allowing edges to be replaced by 'hyperedges'. The cycles of r, s and r s correspond to the hypervertices, hyperedges and hyperfaces of H, which determine the embedding of the underlying (and connected) hypergraph into the surface, and their orders give the type of H, say {k, l, m}. The group G generated by r and s induces a group of automorphisms of this hypergraph, preserving the embedding, and acting transitively on the flags (incident hypervertex-hyperedge pairs) of H. When one of the parameters k, l, m is 2, the hypergraph is a graph, and the hypermap is a regular map. The theory of such objects is well developed, and has been thoroughly explained in [10,11]. Without going into too much detail, we need to make a few basic observations. First, the group G has a presentation of the formand (so) is a finite quotient of the ordinary (k, l, m) triangle group. For simplicity, we will say that such a group G has type (k, l, m), provided that k, l, m are the true orders of the corresponding elements r, s, t. There is a bijective correspondence between isomorphism classes of regular hypermaps of a given type {k, l, m} and torsionfree normal subgroups of the ordinary (k, l, m) triangle group (k, l, m), and the number of those with a given group G as 'rotational symmetry group' (or quotient of (k, l, m)) is equal to the number of ways of generating G by a (k, l, m)-triple (r, s, t) up to equivalence under Aut(G). For further details about representing hypermaps in the form of cellular decomposition of closed two-dimensional surfaces and visualizing the rotational symmetries, and also their association with Riemann surfaces and algebraic number fields (through Grothendieck's theory of dessins d'enfants), we refer the reader to [4,10,11].A regular hypermap may admit a symmetry that induces a reversal of some local orientation of the supporting surface. At the group theory level, this is equivalent to the existence of an automorphism ϑ of a (k, l, m)-group G presented as above, such that ϑ inverts two of the three generators. Such regular hypermaps are called reflexible. If ϑ is actually given by conjugation of some element of order two in G, then the corresponding (k, l, m)-generating triple for G gives rise to two distinct reflexible hypermaps: one ...

This is the first in the series of articles stemming from a systematical investigation of finite edge-transitive tetravalent graphs, undertaken recently by the authors. In this article, we study a special but important case in which the girth of such graphs is at most 4. In particular, we show that, except for a single arc-transitive graph on 14 vertices, every edge-transitive tetravalent graph of girth at most 4 is the skeleton of an arctransitive map on the torus or has one of these two properties:(1) there exist two vertices sharing the same neighbourhood, (2) every edge belongs to exactly one girth cycle.Graphs with property (1) or (2) are then studied further. It is shown that they all arise either as subdivided doubles of smaller arc-transitive tetravalent graphs, or as line graphs of triangle-free (G, 1)-regular and (G, 2)-arc-transitive cubic graphs, or as partial line graphs of certain cycle decompositions of smaller tetravalent graphs.