Schur's exponent conjecture - counterexamples of exponent ۵ and exponent ۹

There is a long-standing conjecture attributed to I. Schur that if G is a finite group with Schur multiplier M(G) then the exponent of M(G) divides the exponent of G. In this note I give an example of a four generator group G of order ۵^{۴۱۲۲} with exponent ۵, where the Schur multiplier M(G) has exponent ۲۵.

Keywords:

Schur multiplier , Groups of exponent ۵ , Schur’s exponent conjecture