Although we have defined a group abstractly, they were not always defined in this way. When Galois introduced the term groupe, he only referred to a subset of permutations that was closed under multiplication. Hence, he only was considering the subgroups of a special type of group, known as permutation groups. However, with these permutation groups, he Was able to prove that most fifth-degree polynomials cannot be solved in terms of roots. Hence, permutation groups have historically been at the core of abstract algebra.