Let S denote the class of all
finite groups all of whose irreducible characters over C (the complex numbers) are
real. It is easy to verify, but important to observe, that this condition is
equivalent to the condition that every element of the group is conjugate to its
inverse (under an inner automorphism). Since Sn∈ S, where Sn denotes the
symmetric group on n letters, any finite group may be embedded in a group
in S. The goal of §1 will be to show that the two-Sylow subgroups of Sn
also are in S, and, if An denotes the alternating group on n letters, that
An∈ S if and only if n ∈{1,2,5,6,10,14}. The result on the two-Sylow
subgroups of Sn will be used to show that any finite two-group is embeddable
in a two-group in S. G. A. Miller has studied a class of groups related to
those in S. The main theorem of §2 gives a more intuitive characterization
of the class of groups investigated by Miller, a consequence of which is a
necessary and sufficient condition for a group in this class to be a member of
S.
