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abstractly Bumside Burnside Cole and Glover congruences inconsistent corre corresponds a single corresponds the product defining relations elements of G exponents find the number formulae hold four operations further conditions G and G group G group of isomorphisms group of order Hence hold for groups independence of P0 insure that P0 insure the independence Journal of Mathematics Let P0 Let P0=Px Let x1 MARRIOTT mod p2 modp morphisms non-abelian group number of elements number of solutions operation of G operations of order order p2 P0 and Q0 product PQ Px Q Q to Q0 Q-iPQ Q-lPQ Q0 and R0 Ql=l QV px R-iQR R-lPR R-lPR=P R0 are independent Rp=l satisfy Section 15 Sections 11 set of congruences simply isomorphic single operation sponds sufficient to insure Suppose P0Q0 tabulation Theory of Groups transformation formulae University of Pennsylvania x—x*z+z x—xy+y x$0 mod