Unformatted text preview: 210 (b)
(c) 3. PRODUCTS OF GROUPS Find the explicit decomposition Œ24 D a2 C a3 , where aj 2
AŒj .
Find the unique x satisfying 0 Ä x Ä 35, with x Á 3 .mod 4/,
x Á 6 .mod 9/. 3.6.12. Consider Z180 . Note that 180 D 4 9 5.
(a) Give the explicit primary decomposition of Z180 ,
Z180 D AŒ2 AŒ3 AŒ5: Find the explicit decomposition Œ24 D a2 C a3 C a5 , where
aj 2 AŒj .
(c) Find the unique x satisfying 0 Ä x Ä 179, with x Á 3 .mod 4/,
x Á 6 .mod 9/, and x Á 4 .mod 5/.
In order to do the computations in part (b), you will need to ﬁnd integers
t2 ; t3 ; t5 such that t2 45 C t3 20 C t5 36 D 1.
(b) 3.6.13. Prove the uniqueness statement in the Chinese remainder theorem,
Theorem 1.7.9.
3.6.14. Show that .Z10 Z6 /=A Š Z2 Z3 , where A is the cyclic subgroup of Z10 Z6 generated by .Œ210 ; Œ36 /.
3.6.15. How many abelian groups are there of order p 5 q 4 , where p and q
are distinct primes?
3.6.16. Show that Za Zb is not cyclic if g:c:d:.a; b/ 2. 3.6.17. Let G be a ﬁnite abelian group, let p1 ; : : : ; pk be the primes dividing jG j. For b 2 G , write b D b1 C C bk , where bi 2 GŒpi . Show
Q
that o.b/ D i o.bi /:
3.6.18. Suppose a ﬁnite abelian group G has invariant factors
.m1 ; m2 ; : : : ; mk /. Show that G has an element of order s if, and only
if, s divides m1 .
3.6.19. Find the structure of the group ˚.n/ for n Ä 20. ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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