GROUP THEORY
(a) Identify the distinct relevant axes and count the number of qualifying rota-
tions about each.
(b) The orientation of the cube is determined if the directions of two of its body
diagonals are given. Consider the number of distinct ways in which one body
diagonal can be chosen to be ‘vertical’, say, and a second diagonal made to
lie along a particular direction.28.11 Identify the eight symmetry operations on a square. Show that they form a
groupD 4 (known to crystallographers as 4mmandtochemistsasC 4 v) having one
element of order 1, five of order 2 and two of order 4. Find its proper subgroups
and the corresponding cosets.
28.12 IfAandBare two groups, then their direct product,A×B,isdefinedtobe
the set of ordered pairs (X, Y), withXan element ofA,Yan element ofB
and multiplication given by (X, Y)(X′,Y′)=(XX′,YY′).Prove thatA×Bis a
group.
Denote the cyclic group of ordernbyCnand the symmetry group of a regular
n-sided figure (ann-gon) byDn– thusD 3 is the symmetry group of an equilateral
triangle, as discussed in the text.
(a) By considering the orders of each of their elements, show (i) thatC 2 ×C 3 is
isomorphic toC 6 , and (ii) thatC 2 ×D 3 is isomorphic toD 6.
(b) Are any ofD 4 ,C 8 ,C 2 ×C 4 ,C 2 ×C 2 ×C 2 isomorphic?
28.13 Find the groupGgenerated under matrix multiplication by the matrices
A=
(
01
10
)
, B=
(
0 i
i 0)
.
Determine its proper subgroups, and verify for each of them that its cosets
exhaustG.
28.14 Show that ifpis prime then the set of rational number pairs (a, b), excluding
(0,0), with multiplication defined by
(a, b)•(c, d)=(e, f), where (a+b√
p)(c+d√
p)=e+f√
p,
forms an Abelian group. Show further that the mapping (a, b)→(a,−b)isan
automorphism.
28.15 Consider the following mappings between a permutation group and a cyclic
group.
(a) Denote byAnthe subset of the permutation groupSnthat contains all the
even permutations. Show thatAnis a subgroup ofSn.
(b) List the elements ofS 3 in cycle notation and identify the subgroupA 3.
(c) For each elementXofS 3 ,letp(X)=1ifXbelongs toA 3 andp(X)=−1ifit
does not. Denote byC 2 the multiplicative cyclic group of order 2. Determine
the images of each of the elements ofS 3 for the following four mappings:
Φ 1 :S 3 →C 2 X→p(X),
Φ 2 :S 3 →C 2 X→−p(X),
Φ 3 :S 3 →A 3 X→X^2 ,
Φ 4 :S 3 →S 3 X→X^3.(d) For each mapping, determine whether the kernelKis a subgroup ofS 3 and,
if so, whether the mapping is a homomorphism.28.16 For the groupGwith multiplication table 28.8 and proper subgroupH={I, A,B},
denote the coset{I, A,B}byC 1 and the coset{C, D, E}byC 2 .Formthesetof
all possible products of a member ofC 1 with itself, and denote this byC 1 C 1.