By Choudhary P.

ISBN-10: 8189473549

ISBN-13: 9788189473549

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**Extra info for Abstract algebra**

**Example text**

Proof: Let f g E X-L4/}/e j and seek to show / and g commute. For all i E /, /(/) and g(i) belong to A f so f(i)g (i) = g (i)f(i). 3 I f -L4Z} Z(E/ is an indexed set o f reduced abelian groups, then the direct product X{v4z} zG/ is reduced. Proof: Let D be a divisible subgroup o f X-L4/}/<=/ and seek to show that D is trivial. Let GD. By Theorem A . 8. 1, 7rz[Z>] is divisible for all i E Z But it( [D] is a subgroup o f the reduced group A t so 717[Z>] is trivial for all i E Z Therefore d(i) = Ki(d) = 0 for all i, hence d = 0 as required.

Then for 1 < k < n, there exists wk E A ; such that 0/ (wk) = h(i k) by the surjectivity assumption. D efin e/by i f z = ik for some k, otherwise. Clearly/has finite support so belongs to 2*L4I-D-,-e/. Then 0(/)(z) = Thus 0 (/) and h agree on all z E If hence are equal completing the proof. 11 Let I and K be equivalent sets. Let A be a group, and let Ai = A for all i E / ¿7«*
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*Semidivisible 3. perfect 4. Let (A, { Z f|}z*ejv> -C/iT/eiv) be a countable central pushout system and let Z be its central pushout. I f each H( is solvable (various solvability lengths al lowed), show Z is hyperabelian. 7 Restricted Wreath Products The wreath product is a somewhat complicated but enormously useful con struction which has a legion o f applications to both finite and infinite group the ory. Actually there are two wreath product constructions. The one to be consid ered in this section is based on restricted direct products. *

### Abstract algebra by Choudhary P.

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