Vol. 5 No 10 - Pi Mu Epsilon

a7r ~[p"], there exists an element a E G[pn] such that hG(x + a) < hG(x + z).Consequen:ly for each positive integer N there exists a e G[pn] such thath(x + a ) > N. If a E G[pn] then pn(x + a) = p x and hG(pnx) =hG(pn(x + a)) 2 hG(x + a).Thus hG(6'x) 7 hG(x + a) for each a E G[pE]and consequently hG(pnx) > N for each positive integer N.pnx f 0 and hG(pnx) > ID,Therefrrecontradictory to the assumption that A(G) = ID.Thus if A(G) 5 ID, G[pn] is a nice subgroup of the reduced p-group G.A topology may be introduced on a reduced p-group G by letting thesubgroups {pn~},z