Realizability of representations of finite groups

A complex character of a finite group G is called orthogonal if it is the character of a real representation. If all characters of G are orthogonal, then G is called totally orthogonal. Totally orthogonal groups are generated by involutions. Necessary and sufficient conditions for total orthogonality are obtained for 2-groups, for split extensions of elementary abelian 2-groups, for Frobenius groups, and for groups whose irreducible character degrees are bounded by 2. Sylow 2-subgroups of alternating groups and finite reflection groups are observed to be totally orthogonal.