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1905.] THE CONCEPT OF INFINITY. 499 of the orders 4 and 8. Hence, there are just four groups in which H is necessarily non-abelian. In two of these GjA is cyclic while the quotient group is non-cyclic in the other two. The total number of the non-abelian groups of order 2 m which contain an invariant cyclic subgroup of order 2 m~2 , but no such subgroup of order 2 m_1 is therefore fourteen. The last four were explicitly excluded from my list of these groups which do not contain an abeliandoi:10.1090/s0002-9904-1905-01253-2 fatcat:3rybzo3xcbbodf5nhg3ka5rmka