Orthomorphism graphs of groups.

*(English)*Zbl 0796.05001
Lecture Notes in Mathematics. 1535. Berlin: Springer-Verlag. viii, 114 p. (1992).

A permutation \(\varphi\) of a finite group \(G\) is called an orthomorphism if the mapping \(x \to x^{-1} \varphi (x)\) is again a permutation. Two orthomorphisms \(\varphi\) and \(\tau\) are adjacent if \(x \to \varphi (x)^{-1} \tau (x)\) is a permutation of \(G\). The set of orthomorphisms together with this adjacency relation is the orthomorphism graph. Orthomorphisms are related, for instance, to affine planes, nets, Latin squares, relative difference sets and Hadamard matrices. The major question is the classification of groups admitting orthomorphisms. The famous Hall-Paige Theorem asserts that a group with a nontrivial cyclic Sylow 2-subgroup cannot admit an orthomorphism. Other interesting problems are the construction of maximal (not extendable) sets of mutually orthogonal orthomorphisms (which are cliques in the orthomorphism graph). It is easy to see that there are at most \(| G|-2\) mutually adjacent orthomorhpisms in \(G\). Sets of this size are called complete and they correspond to projective planes which are \((p,L)\)-transitive for some incident point-line pair \((p,L)\). In general, cliques in the orthomorhpism graph correspond to certain nets. The methods to study orthomorphisms are algebraic. From graph theory, only the terminology of a clique is used.

Following the introductory chapter, Evans describes orthomorphisms in elementary-abelian groups. Then he studies the orthomorphism graph of the multiplicative group of a finite field. The next chapter contains results about the orthomorphism graph generated by automorphisms. In this case, mutually orthogonal orthomorphisms describe translation nets. The book also contains a list of orthomorphisms in some small groups.

Many results about orthomorphisms are scattered throughout the literature in very different terminology. Evans’s monograph is a unifying approach and a nice introduction to this interesting area of Discrete Mathematics. The monograph under review summarizes basically all known results about this topic. The presentation is well organized. The book does not only develop the theory nicely but also contains a lot of open problems which might stimulate the future research.

The book is accessible for students with some algebraic background.

Following the introductory chapter, Evans describes orthomorphisms in elementary-abelian groups. Then he studies the orthomorphism graph of the multiplicative group of a finite field. The next chapter contains results about the orthomorphism graph generated by automorphisms. In this case, mutually orthogonal orthomorphisms describe translation nets. The book also contains a list of orthomorphisms in some small groups.

Many results about orthomorphisms are scattered throughout the literature in very different terminology. Evans’s monograph is a unifying approach and a nice introduction to this interesting area of Discrete Mathematics. The monograph under review summarizes basically all known results about this topic. The presentation is well organized. The book does not only develop the theory nicely but also contains a lot of open problems which might stimulate the future research.

The book is accessible for students with some algebraic background.

Reviewer: A.Pott (Augsburg)

##### MSC:

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

51E14 | Finite partial geometries (general), nets, partial spreads |

11T22 | Cyclotomy |

20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |

51E15 | Finite affine and projective planes (geometric aspects) |

05B15 | Orthogonal arrays, Latin squares, Room squares |