THE UNIVERSITY OF SYDNEY
Math29688 Algebra (Advanced)
Semester 2
Tutorial Solutions Week 2
2008
1.
(for general discussion) Explain how the conjugation principle applies in the following situations:
(a) using a bus to travel from point A to point B;
(b) using an automatic telling machine;
(c) taking a cake out of the oven (careful, hot!);
(d) the survival of a chameleon.
Solution
(a) Perform
XYX
−
1
where
X
is the operation of stepping on the bus and
Y
is the operation of
remaining seated while the bus moves from
A
to
B
.
(b) Perform
XYX
−
1
where
X
is the operation of putting your card into the slot and
Y
is the operation
of typing the various keys of the machine.
(c) Perform
X
1
X
2
YX
−
1
2
X
−
1
1
where
X
1
is the operation of putting on oven mits,
X
2
is the operation
of opening the oven door and
Y
is the operation of taking the cake from the open oven to the kitchen
bench. Note that
X
1
and
X
2
and their inverses can be performed in either order.
(d) The chameleon survives by camouflage, so, assuming it starts and finishes in an environment of the
same base colour, a portion of its life may be modelled by a sequence of the form
X
1
Y
1
X
−
1
1
X
2
Y
2
X
−
1
2
...X
k
−
1
Y
k
−
1
X
−
1
k
−
1
X
k
Y
k
X
−
1
k
where
X
i
is the operation of changing from a base colour to colour
i
and
Y
i
is the operation of living
whilst in the environment of colour
i
.
A more accurate model would replace each
X
−
1
i
X
i
+1
by the
operation of changing directly from colour
i
to colour
i
+ 1.
2.
Express the following permutations of
{
1
,
2
,
3
,
4
,
5
,
6
}
using cycle notation:
α
: 1
mapsto→
3
,
2
mapsto→
5
,
3
mapsto→
6
,
4
mapsto→
2
,
5
mapsto→
1
,
6
mapsto→
4
β
: 1
mapsto→
2
,
2
mapsto→
1
,
3
mapsto→
3
,
4
mapsto→
6
,
5
mapsto→
5
,
6
mapsto→
4
γ
: 1
mapsto→
4
,
2
mapsto→
3
,
3
mapsto→
6
,
4
mapsto→
5
,
5
mapsto→
1
,
6
mapsto→
2
Solution
We have
α
= (136425),
β
= (12)(46) and
γ
= (145)(236).
3.
Find the composites
αβ
,
αγ
and
βγ
where
α
,
β
and
γ
are the permutations defined in the previous
question. Express the inverses of
α
,
β
,
γ
as positive powers of
α
,
β
,
γ
respectively.
Solution
We have
αβ
= (134)(25),
αγ
= (165432),
βγ
= (1365)(24),
α
−
1
=
α
5
,
β
−
1
=
β
and
γ
−
1
=
γ
2
.
4.
Let
α
: 1
mapsto→
2
mapsto→
3
mapsto→
1 and
β
: 1
mapsto→
1
,
2
mapsto→
3
mapsto→
2. Verify the following equations (where 1 denotes the
identity permutation):
α
3
=
β
2
= 1
, βα
=
α
2
β, βαβ
=
α
−
1
.
Put
G
=
(
α,β
)
, the group generated by
α
and
β
. Deduce that
G
=
{
α
i
β
j

0
≤
i
≤
2
,
0
≤
j
≤
1
}
.
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If the letters 1
,
2
,
3 are vertices of an equilateral triangle, interpret elements of
G
in terms of their
geometric effects on the triangle. How many different interpretations of the symbol 1 appear in this
exercise? Are you bothered by this?
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 One '09
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 Algebra, Bijection, x1 x2, X2 Y2 X2, X2 Y X2

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