Exponent-Critical Groups

We define and investigate the property of being `exponent-critical' for a finite group. A finite group is said to be exponent-critical if its exponent is not the least common multiple of the exponents of its proper non-abelian subgroups. Recent computational work about varieties of groups relies on the fact that the groups in question are not exponent-critical. In this paper we explore properties of exponent-critical groups and give a characterization of such groups.