## Infinite Abelian groups by Laszlo Fuchs

By Laszlo Fuchs

Fuchs L. endless Abelian teams, vol.2 (AP, 1973)(ISBN 0122696026)

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The contents of this e-book were used in classes given through the writer. the 1st used to be a one-semester path for seniors on the college of British Columbia; it was once transparent that strong undergraduates have been completely in a position to dealing with straightforward team thought and its software to basic quantum chemical difficulties.

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Thus + @ [ + ( A ) n (Ci/Ei)l, I and afortiori + ( A ) , is essential in M / E . Since + ( A ) has no proper essential extension, + ( A ) = M / E , whence M = A 0 E , in fact. Next let A = @ Z ( p " ) with fixed p". Assume ( I ) with the C i satisfying p"Ci = 0. The argument in the preceding paragraph can be repeated, except for the last sentence. Instead, one should observe that p " ( M / E ) = 0 implies, in view ofthe structure of A , that M / E can not be a proper essential extension of + ( A ) 2 A .

B, and hence on B. , / ? - ‘ a x , = P ’ - ’ a ’ x i . Thus #J is a well-defined homomorphism B - + A . u = 0, ax = 0, and x = 0, that is, #J is monic. Therefore, #JE is a subgroup of A such that #JB= B. 4)(iv) implies that #JE is a direct summand of A . 0 s, -+ #J #Js 69. 23 TORSION-COMPLETEp-GROUPS If we confine our attention to separable p-groups, the last theorem can be improved. 3 (Leptin [3]). Suppose A is a separable p-group and B is a basic subgroup of A such that every automorphism of B is extendible to an automorphism of A .

Let A be a torsion-complete p-group, and assume (1) with the C i isomorphic to subgroups of A . 2) implies the existence of an integer m such that p r n A [ p ]5 c, 0 . 0 Ck = C' [finite direct sum]. 1), we can leave A , out of consideration. Clearly, A , is torsion-complete, and therefore so is n A , E A , , where n denotes the obvious projection of M onto C'. Because of the purity of A , in M, it is readily verified that n A , is 72. 35 THE EXCHANGE PROPERTY pure in C', whence C' = zrA, 0 N' for some N' C'.