Real elements and real-valued characters of covering groups of elementary abelian 2-groups

In Section 2 of this paper, the maximum number of real elements possible in a covering group of C2(n) is determined, and a description of those covering groups in which this maximum is attained is given. Among these “maximally real” examples is that covering group G of C2(n) which is generated by n involutions. For this particular group, the Schur indices of real-valued irreducible characters of each degree are investigated in Section 4. The main result of this section is a set of recurrence relations describing the number of absolutely irreducible characters of G of a given degree of each of three types (non-real-valued and real-valued of index 1 or 2 over ) in terms of related numbers for the corresponding group on n−1 generators.