Math 653
Homework #4
September 25, 2012
Due Thursday, October 4.
Justify all of your work.
Problem D1: Consider the groups
Y
A=
Zp ,
B=
p prime
M
Zp ,
p prime
where the direct product and direct sum are indexed over all prime numbers p.
(a) Show that every element of B has finite order.
(b) Show that A contains at least one element of infinite order.
(c) Let C = {a ∈ A : |a| < ∞}. Does C = B?
Problem D2: For an abelian group A (written additively) and a positive integer n, let
A[n] = {a ∈ A | n · a = 0},
which is a subgroup of A. Now fix a prime number p, and define
(
)
∞
Y
i i−j
A[p ] ∀ i ≥ j ≥ 1, p · ai = aj .
Rp (A) = (ai ) ∈
i=1
Also, define
(
Sp (A) =
(bi ) ∈
∞
Y
i=1
)
j
A/p A ∀ i ≥ j ≥ 1, bi ≡ bj (mod p A) .
i
(a) Show that Sp (Z) is torsion free. That is, show that every non-identity element of Sp (Z) has
infinite order.
(b) Show that Rp (Q/Z) ∼
= Sp (Z).
(c) Show that Rp (Q/Z) ∼
= Rp (C× ). (Part of the difficulty here is that the multiplicative group
×
C is not usually written additively.)
Problem D3: Consider the subgroup G of GL2 (C) generated by
ω 0
0 i
and
,
0 ω2
i 0
√
√
where i = −1 and ω = e2πi/3 = − 12 + 23 i.
(a) Show that G has order 12.
(b) Show that G is not isomorphic to D6 or A4 .
(c) Show that G ∼
= a, b | a6 = 1, b2 = a3 = (ab)2 .
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