336A2SOLNS.pdf - PMATH 336 Due: June 9, 2020 at 9:00 AM...

PMATH 336 Due: June 9, 2020 at 9:00 AM
Assignment 2 Comments
1. (Xingchi) Let I be the index set of nonzero real numbers and for every α ∈ I let G
α
be the group of nonzero complex numbers equipped with the operation a ? b = αab .
Consider
G =
Y
α ∈ I
G
α
.
(a) What is the identity of G ?
(b) Consider f ∈ G given by f ( α ) = α + i . What is f
- 1
?
(c) Let f be as above and let g ∈ G be given by g ( α ) = 1 + αi . Compute fg .
2. (Aiden)
(a) Let I be a nonempty index set and for every i ∈ I let G
i
be a group. Suppose
H
i
≤ G
i
for every i ∈ I . Prove that
Y
i ∈ I
H
i
≤
Y
i ∈ I
G
i
.
1

PMATH 336 Due: June 9, 2020 at 9:00 AM
Proof. Let e
i
denote the identity of G
i
. First, the identity f ( i ) = e
i
∈ H
i
is an element of
Q
H
i
. Now suppose f, g ∈
Q
H
i
so that f ( i ) , g ( i ) ∈ H
i
for
every i ∈ I . Then, f ( i ) g ( i )
- 1
∈ H
i
for every i ∈ I , since H
i
≤ G
i
. Therefore
fg
- 1
∈
Q
H
i
, as required by the subgroup test.
(b) Prove or Disprove: Let G be a group. Every subgroup of G × G is of the form
H × K , where H ≤ G and K ≤ G .