=, + G2 + * *= En Gp, I H = H1+ H2 + * = ZHP Gp Gq = Ox Hp Hq = O (p 5 iq) where the notation is not to be taken to mean that the sets are denumerable; the elements of a set Gp will be denoted by gp, gp, .. . . Then G is said to be homomorphic to H, G H, if (1) gpgq =gr D HpHq _ Hr. If gg = q,, then 1H1H, H.; but Hr H., = 0 if r $ s; hence r = s, that is, (1) implies (2) GpGq ? Gr. Again, if hahhb = h, let gagb = gd ; then HaHb 5 Hd . But HaHb > he < H,, therefore H, Hd 0 0 so that d = c and, since 9d < Gdi no matter what elements ga and gb are in Ga and Gb, we have hahb = hc D GaGb _ Gc that is, the relation of homomorphism is reflexive. Suppose now that the identity g1 is in G1 ; then gigp = gp = g1yg1 D GGp = Gp = GpG1 for all p, and therefore (3) H1Hp _ Hp I Hp)71 < Hp. If the identity of H lies in Ha, it follows similarly that HaHp = Hp = HpHa and in particular HaHi = H1 = HiHa But from (3) HaHi _ Ha and hence a = 1, so that the identity of H lies in H1. Since G is a group, any gl has an inverse, say gp, such that