Transversals in Groups with an application to loops
Final Report Abstract
Let G be a group and H a subgroup of G. A (right) transversal for H in G is a set of representatives of the set of right cosets of H in G. The aim of the project was to study transversals of subgroups in a finite group G. We investigated mainly the following kind of questions: Suppose a certain type of transversal exists - for instance a normal or global transversal - for a subgroup of G, or for each subgroup of G or a special class of subgroups of G. What is the implication on the structure of G? We focused primarily on the question to determine the structure of the finite groups G which satisfy the following: Hypothesis (A) Assume that G has a subgroup H such that there is a transversal K to H in G which is the union of 1 ∈ G and G-conjugacy classes of involutions. It has been conjectured by several people Conjecture 8.1 If (G, H, K) satisfies Hypothesis (A) and if G is generated by K, then G is a 2-group. Transversals are also extensively studied in finite geometry and since some time in computer science, see for instance [22, 12, 13]. In semigroup theory as well as in geometry they appear in the language of loops. A set X together with a binary operation ◦ on X is a right loop if X contains an identity 1 and if for each x in X the maps R(x) and L(x) on X defined by R(x)(y) = y ◦ x and L(x)(y) = x ◦ y for all y in X are permutations of X. Baer has shown that given a transversal then we can define a multiplication on it and thereby get a right loop. Moreover, every right loop is obtained in this way. Notice that a loop can be thought of as a group without associativity law. The groups G which satisfy Hypothesis (A) are precisely those which are related to so called Bol loops of exponent 2. Every such loop is also a Bruck loop. We managed to disprove Conjecture 8.1 by presenting a counterexample. This then gave the impression that a group related to a Bol loop of exponent 2 or to a Bruck loop could be very wild. We could completely describe the structure of the groups related to finite Bruck loops. This structure turned out to be surprisingly easy. Moreover, we managed to show generalisations of classical group theoretic theorems such as Sylow’s Theorem, Lagrange’s Theorem and Hall’s Theorem to Bruck loops. We also obtained results concerning global groups and normal transversals. In addition we continued the classification of finite primitive permutation groups which possess a regular subgroup. This became a joint project with Praeger (Perth, Australia).
Publications
- On Bruck loops of 2-power-exponent II (2009), 30 Seiten
A. Stein
- Commuting graphs of odd order elements in simple groups (2010), 26 Seiten
B. Baumeister, A. Stein
- On Bruck loops of 2-power-exponent (2010), 26 Seiten
B. Baumeister, A. Stein, G. Stroth
- Self-invariant 1-Factorizations of complete graphs and finite Bol Loops of Exponent 2, Beiträge zur Algebra und Geometrie 51 (2010), 117 – 135
B. Baumeister, A. Stein
- The finite Bruck loops (2010), 17 Seiten
B. Baumeister, A. Stein
