College Algebra Exam Review 199

# College Algebra Exam Review 199 - 3.6 FINITELY GENERATED...

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Unformatted text preview: 3.6. FINITELY GENERATED ABELIAN GROUPS 209 Another way to ﬁnish the proof of part (c) is to observe that hx i \ hy i D f1g, since the orders of these cyclic subgroups are relatively prime. It follows then that the subgroup generated by x and y is the direct product hx i hy i Š Zp 1 Zpn 1 Š Zpn 1 .p 1/ . I Remark 3.6.29. All of the isomorphisms here are explicit, as long as we are able to ﬁnd a generator for ˚.p/ for all primes p appearing in the decompositions. Exercises 3.6 3.6.1. Let G be an abelian group. Show that Gtor is a subgroup and G=Gtor is torsion free 3.6.2. Let G be an abelian group. Suppose that G D A B , where A is a torsion group and B is free abelian. Show that A D Gtor . 3.6.3. Let B be a maximal linearly independent subset of an abelian group G . Show that ZB is free and that G=ZB is a torsion group. 3.6.4. Show that an abelian group is ﬁnite if, and only if, it is a ﬁnitely generated torsion group. 3.6.5. Let G be a ﬁnite abelian group. Show that ann.G/ D fr 2 Z W rx D 0 for all x 2 G g is a nonzero subgroup of Z. Show that ann.G/ D aZ, where a is the smallest positive element of ann.G/. 3.6.6. Suppose where bj bt . G Š Zb1 Zb2 Zb t ; 2, and bi divides bj for i Ä j . Show that the period of G is 3.6.7. Find the elementary divisor decomposition and the invariant factor decomposition of Z108 Z144 Z9 . 3.6.8. Find all abelian groups of order 108. For each group, ﬁnd the elementary divisor decomposition, and the invariant factor decomposition. 3.6.9. Find all abelian groups of order 144. For each group, ﬁnd the elementary divisor decomposition, and the invariant factor decomposition. 3.6.10. How many abelian groups are there of order 128, up to isomorphism? 3.6.11. Consider Z36 . Note that 36 D 4 9. (a) Give the explicit primary decomposition of Z36 , Z36 D AŒ2 AŒ3: ...
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