  A skew Hadamard difference set (SHDS) is a difference set that satisfies the
skew condition. It is known that if a group G admits a skew hadamard difference
set, then G is a p-group with order congruent to 3 modulo 4. We will generalize
skew Hadamard difference sets to Generalized Skew Hadamard Difference Sets
(GSHDS) to cover the case when p is congruent to 1 module 4, and we will extend
all known results that yield necessary existence conditions of skew Hadamard
difference sets to our generalization, including the known exponent bounds. We
will also show a set of necessary existence conditions a special family of
groups. We will close the article with a general p-divisibility condition of
the difference intersection numbers of a GSHDS for a special class of subgroups
L.
