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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 Fall '08
 EVERAGE
 Algebra, Abelian group, Cyclic group, elementary divisor decomposition

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