By Paul S. Alexandroff

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Knowing S2 = e and st = t-is allows us to multiply any two elements from the list (**) and manipulate the product to have the same form. This reminds us very much of D n • Indeed, the only difference is that the rotation r of order n has been replaced by the translation t of infinite order. For this reason we call G the infinite dihedralgroup and denote it by Deo. We end this section with one or two useful facts about subgroups. 1) Theorem. A non-empty subset Hof a group G is a subgroup ofG ijand only ijxy-i belongs to H whenever x and y belong to H.

Sinee any subgroup of an abelian group is abelian, we see that if G x His abelian, then so are both G and H. The direet produet GI x ... x Gn of a finite eolleetion of groups has elements (Xl' ••. ,Xn ) where Xi E Gi' I ~ i ~ n, whieh are eombined via (Xl' . • , Xn)(X~, • •. , X~) = (XIX~, ••. , XnX~). Again, ehanging the order of the faetors always produees an isomorphie group. EXAMPLES. (i) Z2 x Z3 has six elements, (0,0), (1,0), (0, 1), (1, 1), (0,2), (1,2), whieh are eombined by (x,y) + (x',y') = (x +2 x',y +3 y').

8. Which elements of the infinite dihedral group have finite order? 00 these elements form a subgroup of Doo ? 9. LetJ be a function from the realline to itselfwhich preserves the distance between every pair of points and which sends the integers among themselves. (a) AssumingJhas no fixed points, show thatJis a translation through an integral distance. (b) IfJleaves exactly one point fixed, show that this point is either an integer or lies midway between two integers, and thatJis reflection in this fixed point.