TITLE: Which groups have only real representations?
QUESTION [7 upvotes]: An irreducible representation $\rho$ (with character $\chi$) of a finite group is called a "real" representation if its Frobenius-Schur indicator is 1:
$$\frac{1}{\lvert G \rvert} \sum_{g \in G} \chi\left(g^2\right) = 1$$
Alternatively, it is real if it has a symmetric bilinear, $G$-equivariant form.
Finite groups can have real, quaternionic and complex representations. Some groups like the symmetric group have only real representations.
Which groups have only real representations?

REPLY [3 votes]: The correct term for this kind of groups is called "totally orthogonal" (e.g. orthogonal groups have this property). It was coined in this article by Wang and Grove, together with many propositions about their structure.
For example, it contains:
Theorem 3.3 If $G$ is totally orthogonal, it is generated by involutions.