Ph.D Thesis
Ph.D Student Sambonet Nicola On the Exponent of The Schur Multiplier Department of Mathematics Professor Eli Aljadeff Dr. Yuval Ginosar

Abstract

The Schur multiplier is a very interesting invariant, it has been introduced in the beginning of the twentieth century by Issai Schur, aimed at the study of projective representations of finite groups. After several decades, the multiplier became the archetype for groups cohomology. Nowadays this subject is covered by a very dense theory, nonetheless it merit further investigation. An instance is given by the fact that, given a group G, an explicit description of the Schur multiplier M(G) is often a too difficult task. Therefore, it is of interest to describe bounds for the arithmetical properties of the multiplier, as the order, the rank, and - our subject - the exponent.

In his seminal work Schur already showed that exp M(G) divides the order of the group, and that for abelian groups it divides the exponent of the group. It has been a long standing conjecture that this property holds for any group. The validity of the conjecture has been confirmed for many families of groups, nevertheless the general validity of the conjecture has been disproved. Still the counterexamples we know are essentially 2-groups, and the problem remains open for groups of

odd order. The natural setting for the theory of the Schur multiplier is indeed represented by the p-groups. There are many results in this direction, where bounds are given for specific classes of p-groups. For instance, bounds for exp M(G) were described in terms of the nilpotency class, the derived length, the rank, and the coclass.

With the present research thesis we have contributed to improve these bounds, introducing a new construction based on the original work of Schur but precisely aimed at the exponent problem. A classical theorem proves for any given group the existence of at least one Schur cover, which are particular central extensions of minimal order encoding all of the information contained in the multiplier. Nevertheless, there are other extensions which minimize the exponent in place of the

order. This phenomenon is counterintuitive as in order to appreciate it there is a need to consider also some coboundaries which are usually regarded as superfluous. On the other hand, the unitary cover Γu(G) - our construction - is defined in a very intuitive way, requiring only that the section of G in Γu(G) preserves the order of the elements.

Beside to give a better control of the exponent of the Schur multiplier, the unitary cover behaves very well with respect to subgroups and quotients. Consequently, it has been possible to improve the bounds in terms of the invariants of the group, and to confirm the original conjecture for new families of groups. Moreover, the good behavior with respect to quotients reduces the problem to the use of the Kostrikin-Zelmanov solution of the restricted Burnside problem for 2-generator groups of exponent p^k.