College Algebra Exam Review 95

# College Algebra Exam Review 95 - 2.3 THE DIHEDRAL GROUPS...

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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 2.2.34.) 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 computer language: n D 8; For[j D 1, j Ä 2n 2.2.30. (a) (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 disk 23 x 4y 5 W x 2 C y 2 Ä 1g; f 0 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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## This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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