Math 5125
Friday, November 18
November 18, Ungraded Homework
Exercise 10.5.1 on page 403
Suppose that
A
ψ
→
B
φ
→
C
α
y
β
y
γ
y
A
0
ψ
0
→
B
0
φ
0
→
C
0
is a commutative diagram of groups and that the rows are exact. Prove that
(a) if
φ
,
α
are surjective and
β
is injective, then
γ
is injective.
(b) if
ψ
0
,
α
and
γ
are injective, then
β
is injective,
(c) if
φ
,
α
and
γ
are surjective, then
β
is surjective,
(d) if
β
is injective and
α
,
φ
are surjective, then
γ
is injective,
(e) if
β
is surjective and
γ
,
ψ
0
are injective, then
α
is surjective.
(a) Let
c
∈
ker
γ
. Then
c
=
φ
b
for some
b
∈
B
and 0
=
γ
c
=
γφ
b
=
φ
0
β
b
, so
β
b
∈
ker
φ
0
.
Therefore
β
b
=
ψ
0
a
0
for some
a
0
∈
A
0
, and then we may write
a
0
=
α
a
for some
a
∈
A
because we are given that
α
is surjective. Therefore
βψ
a
=
ψ
0
α
a
=
ψ
0
a
0
=
β
b
,
consequently
ψ
a
=
b
, because we are given that
β
is injective. We conclude that
c
=
φ
b
=
φψ
a
=
0 as required.
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 Fall '07
 PALinnell
 Math, Algebra, γ, β, α, November 18

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