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Math 3124
Thursday, September 22
Fourth Homework Solutions
1.
Problem 8.3 on page 50
Determine the group of symmetries of an equilateral trian
gle.
±
±
±
±
±
±
±
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±
±A
A
A
A
A
A
A
A
A
A
A
A
B
A
C
Let the equilateral triangle be ABC as shown. Any symmetry must permute the ver
tices, and any symmetry which ﬁxes all the vertices must be the identity. Therefore
there are at most 6 symmetries, and we shall see next that there are exactly 6. In
fact we have the identity which we denote by (A), reﬂection in the line through A
to the midpoint of BC which we shall denote (BC), reﬂection in the line through B
to the midpoint of CA which we shall denote (AC), reﬂection in the line through C
to the midpoint of AB which we shall denote (AB), rotation anticlockwise through
120
◦
which we shall denote (ABC), and rotation through 240
◦
which we shall denote
(ACB). Now the composition of mapping is obtained by composing the corresponding
permutations. Therefore the Cayley table is
◦
(A)
(BC)
(CA)
(AB)
(ABC)
(ACB)
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 Fall '08
 PARRY
 Math, Algebra

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