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Unformatted text preview: 2 a 3 b where a = , 1 and b = , 1 , 2 , 3, which is 2 * 4 possibilities. Therefore Z 54 has 8 subgroups; the orders will be 1,3,9,27 and 2,6,18,54. It follows that the subgroup lattice looks like @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ u h [ ] i u h [ 18 ] i u h [ 27 ] i u h [ 9 ] i u h [ 3 ] i u h [ 1 ] i u h [ 6 ] i u h [ 2 ] i The number of subgroups of a cyclic group of order p 2 q 2 Here p , q are distinct primes. There is exactly one subgroup for each integer dividing the order of the group. The numbers dividing p 2 q 2 are p a q b where a = , 1 , 2 and b = , 1 , 2. Therefore the number of subgroups is 3 * 3 = 9....
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 Fall '08
 PARRY
 Math, Algebra

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