We study the generic initial ideals (gin) of certain ideals that arise in
modular invariant theory. For all cases an explicit generating set is known we
calculate the generic initial ideal of the Hilbert ideal of a cyclic group of
prime order for all monomial orders. We also clarify the Klein four group and
note that its Hilbert ideals are Borel fixed with certain orderings of the
variables. In all situations we consider, there is a monomial order such that
the gin of the Hilbert ideal is equal to its initial ideal. Along the way we
show that gin respects a permutation of the variables in the monomial order.Comment: 8 page
