The free product G*H of groups G and H is the set of elements of the form g_1 h_1 g_2 h_2 ...g_r h_r, where g_i element G and h_i element H, with g_1 and h_r possibly equal to e, the identity element of G and H. Free products of more than two groups are defined recursively, i.e., G_1 *G_2 *...*G_n = (G_1 *G_2 *...*G_(n - 1))*G_n. The free group F_n is the free product of Z with itself n times. The notion of free products can be generalized from groups to categories.

category | free group | free semigroup | group