XXXXXXXXXXExercise 3 1 Let G be the symmetry group of the regular pentagon. Find all elements of G...

Question

XXXXXXXXXXExercise 3 1 Let G be the symmetry group of the regular pentagon. Find all elements of G and find generators and relations for G. *2 Let T be a regular tetrahedron, and let G be the symmetry group of T (not allowing reflections). Show that G has 12 elements and draw a diagram to show each of these symmetries. Let s be a rotation of 120o through a line through a vertex and the opposite face, and ? and t be two rotations of 180o through the line joining the midpoint of opposite edges. Show that G = {s a ? b t c : 0 = a = 2, 0 = b = 1, 0 = c = 1} and find generators and relations for G. 3 Find the order of each element of Z10 (under addition) and Z * 7 (under multiplication). 4 (i) Find the inverse of 6 in Z * 11. (Not much calculation needed!) Solve 6x = 9 (mod 11). *(ii) Find the inverse of 33 in Z * 47. (Use the method of finding h, k in the h, k-lemma.) Solve 33x = 20 (mod XXXXXXXXXXWrite down and learn the conditions usually used to prove that a subset H of G is a subgroup. 6 Determine, with justification, whether or not the given sets H are subgroups of the given group G: (i) G = R - {0} under ×, H = {x ? R : x = 1}, (ii) G = R - {0} under ×, H = {2 n : n ? Z}, (iii) G = R - {0} under ×, H = {x ? G : x 2 ? Q}, *(iv) G = GL2(R), H = {A =  a b c d  ? G : ad - bc = 1} *(v) G = GL2(R), H = {A =  a b c d  ? G : a, b, c, d = 0} *(vi) G is any group, H = {g ? G : gx = xg for all x ? G}
	
Document Preview:  XXXXXXXXXXExercise 3
1 Let G be the symmetry group of the regular pentagon. Find all elements of
G and ?nd generators and relations for G.
*2 Let T be a regular tetrahedron, and let G be the symmetry group of T (not
allowing re?ections). Show that G has 12 elements and draw a diagram to show
o
each of these symmetries. Let s be a rotation of 120 through a line through a
o
vertex and the opposite face, and ? and t be two rotations of 180 through the
line joining the midpoint of opposite edges. Show that
a b c
G ={s ? t : 0=a= 2,0=b= 1,0=c= 1}
and ?nd generators and relations for G.
*
3 Find the order of each element of Z (under addition) and Z (under multi-
10
7
plication).
*
4 (i) Find the inverse of 6 in Z . (Not much calculation needed!) Solve 6x= 9
11
(mod 11).
*
*(ii) Find the inverse of 33 in Z . (Use the method of ?nding h,k in the
47
h,k-lemma.) Solve 33x= 20 (mod 47).
5 Write down and learn the conditions usually used to prove that a subset H
of G is a subgroup.
6 Determine, with justi?cation, whether or not the given sets H are subgroups
of the given group G:
(i) G = R-{0} under×, H ={x?R :x= 1},
n
(ii) G = R-{0} under×, H ={2 :n? Z},
2
(iii) G = R-{0} under×, H ={x?G :x ? Q},
 
a b
*(iv) G =GL (R), H ={A = ?G :ad-bc = 1}
2
c d
 
a b
*(v) G =GL (R), H ={A = ?G :a,b,c,d= 0}
2
c d
*(vi) G is any group, H ={g?G :gx =xg for all x?G}
(Note: GL (R) denotes the group of invertible 2×2 matrices with real entries.)
2
You are advised to attempt all questions. Please hand in the assessed ques-
tions (the questions marked with a *) on Monday 18 February at the lecture
immediately after reading week.
1

David · Accepted Answer

1. The symmetries of the regular pentagon is isomorphic to the Dihedral group D5. We can number the
vertices cyclically and we obtain{
1, (1 2 3 4 5), (1 3 5 2 4), (1 4 2 5 3), (1 4 2 5 3), (1 5 4 3 2), (2 5)(3 4), (1 5)(2 4), (1 4)(2 3), (1 3)(4 5), (1 2)(3 5)
}
Since this group is isomorphic to D5 and we know that
D5 = 〈x, y | x5 = y2 = (xy)2 = 1〉
2.
Since rotation is mapped to 3 Cycle by φ and since 3-cycles generate An for n ≥ 3 we have that
φ(Sd(T)) ∼= A4. Note that A4 is the alternating group on 4 symbols, the group of even permutations which
has order |A4| = 4!2 = 12.
3. Z10 = {0̄, 1̄, 2̄, 3̄}. And |1̄| = 4 because 1̄ + 1̄ + 1̄ + 1̄︸ ︷︷ ︸
4 times
= 0, similarly we have |2̄| = 2, since 2̄ + 2̄ = 0, and
|3̄| = 4, since 3̄ + 3̄ + 3̄ + 3̄ = 12 ≡ 0 (mod 4). For Z∗7 is the group which has elements from Z7 such that
1
gcd(a, 7) = 1.

XXXXXXXXXXExercise 3 1 Let G be the symmetry group of the regular pentagon. Find all elements of G and find generators and relations for G. *2 Let T be a regular tetrahedron, and let G be the symmetry...

Solution

Answer To This Question Is Available To Download

Related Questions & Answers

Submit New Assignment