By P. Griffith

ISBN-10: 0226308707

ISBN-13: 9780226308708

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**Example text**

18. Show that E(Z) = III*. Moreover, if C is a reduced countable torP P sion free group, then prove that E(C) is isomorphic to a direct summand of E< I Z>. 19. Let [Cn:n

Show that £a . is an essential subgroup of IB.. 18. £l is a maximal independent subset of a group G such that the order of each x. is infinite or of prime power, then |x| = rank(G). If G is uncountable, show that |x| = |g | = rank(G). 19. If G is Hausdorff and torsion free, prove that § is also torsion free. 20. Prove that a reduced torsion group is algebraically compact if and only if it is bounded. 21. A subgroup A of a group B is called a pure essentia I subgroup if the only subgroup C of B with the properties that A P i C = 0 and A + C pure in B is the zero subgroup.

C) xx_| e Ax and * x_ ( e X x_| if X-l exists. The conditions are trivially met when y = I, X = 0; note that Aq = A and Ao = A. We wish to construct a height-preserving isomorphism that extends ttx , for X < y, in such a way that conditions (a)-(c) are easily verified for X - u. A, = ^ X y A^y A A^>— » Ay and let Ify i s a limit ordinal, define A^ = XL^AX, tt y : A, >— » X be defined by y y tt M = supi^}. X

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