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Math 3124
Thursday, August 25
August 25 Ungraded Homework
Problem 4.1 on page 27
With
S
=
{
a
,
b
}
, the set
M
(
S
)
contains four elements; denote
these by
π
,
ρ
,
σ
,
τ
, deﬁned as follows:
π
(
a
) =
a
ρ
(
a
) =
a
σ
(
a
) =
b
τ
(
a
) =
b
π
(
b
) =
a
ρ
(
b
) =
b
σ
(
b
) =
a
τ
(
b
) =
b
(a) Construct the Cayley table for the composition (
◦
) as an operation on
M
(
S
)=
{
π
,
ρ
,
σ
,
τ
}
.
(As a start,
ρ
◦
τ
=
τ
and
σ
◦
τ
=
π
.)
(b) Which is the identity element?
(c) Is
◦
commutative as an operation on
M
(
S
)
.
(d) Which elements of
M
(
S
)
are invertible?
(e) Is
◦
commutative as an operation on the set of invertible elements in
M
(
S
)
?
The table looks like
◦
π
ρ
σ
τ
π
π
π
π
π
ρ
π
ρ
σ
τ
σ
τ
σ
ρ
π
τ
τ
τ
τ
τ
Some sample calculations: to get
σ
◦
π
=
τ
, we have
σ
◦
π
(
a
) =
σ
(
a
) =
b
σ
◦
π
(
b
) =
σ
(
a
) =
b
,
consequently
σ
◦
π
(
a
) =
τ
(
a
)
for all
a
∈
S
, which proves that
σ
◦
π
=
τ
.
To get
σ
◦
σ
=
ρ
, we have
σ
◦
σ
(
a
) =
σ
(
b
) =
a
σ
◦
σ
(
b
) =
σ
(
a
) =
b
.
From the table, we immediately see that
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This note was uploaded on 01/02/2012 for the course MATH 3124 taught by Professor Parry during the Fall '08 term at Virginia Tech.
 Fall '08
 PARRY
 Math, Algebra

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