Download 1 PROBLEM SET 8 DUE: Apr. 14 Problem 1 Let G, H, K be finitely

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PROBLEM SET 8
DUE: Apr. 14
Problem 1
Let G, H, K be finitely generated abelian groups.
(1). If G ⊕ G = H ⊕ H, then G = H.
(2). If G ⊕ K = H ⊕ K, then G = H.
Problem 2
(1). Use the theorem of elementary divisors give another proof of the
structure theorem of finite generated abelian groups.
(2). Let m1 , m2 , ...., mt and m̃1 , m̃2 , ...., m̃t̃ be two sets of natural numbers with the following properties:
mi |mi+1 , m̃j |m˜j+1 , f or 1 ≤ i ≤ t − 1, 1 ≤ j ≤ t̃ − 1.
Suppose Z/m1 Z ⊕ ..... ⊕ Z/mt Z ∼
= Z/m̃1 Z ⊕ .... ⊕ Z/m̃t̃ Z, show that t = t̃
and mi = m̃i .
(3). For a finite abelian group, provide the transformation law from
the set of invariant divisors to the set of elementary divisors, and vice versa.
(4). Determine all the abelian groups of order 1500 up to isomorphisms, and write down their elementary and invariant divisors.
Problem 3
Show that a finite abelian p-group is generated by its elements of maximal order.
Problem 4
(1). How many subgroups of order p2 does the abelian group Z/p3 Z ⊕
Z/p2 Z have?
(2). Determine the number of cyclic and noncyclic subgroups of order
9 of the group Z3 ⊕ Z9 ⊕ Z9 ⊕ Z243 .
Problem 5
(1). Show that G must be abelian if Aut(G) is of order 2.
(2). Show that Aut(Z2 ⊕ Z2 ) ∼
= S3 .
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(3). Show that any finite group of order ≥ 2 has at least 2 automorphisms.
Problem 6
Let H be a subgroup of a finite abelian group G. Show that G has a
subgroup that is isomorphic to G/H.