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Unformatted text preview: 2.3. THE DIHEDRAL GROUPS 105 2.2.28. Let n
3. Verify that Œ2n 1, Œ2n 1 C 1, and Œ2n 1 1 are
three distinct elements of order 2 in ˚.2n /. Show, moreover, that these are
the only elements of order 2 in ˚.2n /. (These facts are used in Example
2.2.29. Verify that Œ3 has order 2n 2 in ˚.2n / for n D 3; 4; 5. Remark. Using a short computer program, you can quickly verify that Œ3
has order 2n 2 in ˚.2n / for n D 6; 7; 8, and so on as well. For example,
use the following Mathematica program or its equivalent in your favorite
n D 8;
For[j D 1, j Ä 2n
(b) 2, j C C, Print[Mod[3j ; 2n ]]]
k Show that for all natural numbers k , 32
1 is divisible by 2k C2
k C3 . (Compare Exercise 1.9.10.)
but not by 2
Lagrange’s theorem (Theorem 2.5.6) states that the order of any
subgroup of a ﬁnite group divides the order of the group. Consequently, the order of any element of a ﬁnite group divides the
order of the group. Applying this to the group ˚.2n /, we see
that the order of any element is a power of 2. Assuming this,
conclude from part (a) that the order of Œ3 in ˚.2n / is 2n 2 for
all n 3. 2.3. The Dihedral Groups
In this section, we will work out the symmetry groups of regular polygons3
and of the disk, which might be thought of as a “limit” of regular polygons
as the number of sides increases. We regard these ﬁgures as thin plates, capable of rotations in three dimensions. Their symmetry groups are known
collectively as the dihedral groups.
We have already found the symmetry group of the equilateral triangle (regular 3-gon) in Exercise 1.3.1 and of the square (regular 4-gon) in
Sections 1.2 and 1.3. For now, it will be convenient to work ﬁrst with the
4y 5 W x 2 C y 2 Ä 1g;
whose symmetry group we denote by D .
3A regular polygon is one all of whose sides are congruent and all of whose internal angles are congruent. ...
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- Fall '08