 
Summary: LAGRANGE'S THEOREM FOR HOPF MONOIDS IN SPECIES
MARCELO AGUIAR AND AARON LAUVE
Abstract. Following Radford's proof of Lagrange's theorem for pointed Hopf alge
bras, we prove Lagrange's theorem for Hopf monoids in the category of connected
species. As a corollary, we obtain necessary conditions for a given subspecies k of a
Hopf monoid h to be a Hopf submonoid: the quotient of any one of the generating
series of h by the corresponding generating series of k must have nonnegative coeffi
cients. Other corollaries include a necessary condition for a sequence of nonnegative
integers to be the sequence of dimensions of a Hopf monoid in the form of certain
polynomial inequalities, and of a settheoretic Hopf monoid in the form of certain lin
ear inequalities. The latter express that the binomial transform of the sequence must
be nonnegative.
Introduction
Lagrange's theorem states that for any subgroup K of a group H, H = K×Q as (left)
Ksets, where Q = H/K. In particular, if H is finite, then K divides H. Passing to
group algebras over a field k, we have that kH = kK kQ as (left) kKmodules, or
that kH is free as a kKmodule. Kaplansky [7] conjectured that the same statement
holds for Hopf algebrasgroup algebras being principal examples. It turns out that the
result does not not in general, as shown by Oberst and Schneider [16, Proposition 10]
and [14, Example 3.5.2]. On the other hand, the result does hold for certain large
