28.8 EXERCISES
Similarly computeC 2 C 2 ,C 1 C 2 andC 2 C 1. Show that each product coset is equal to
C 1 or toC 2 ,andthata2×2 multiplication table can be formed, demonstrating
thatC 1 andC 2 are themselves the elements of a group of order 2. A subgroup
likeHwhose cosets themselves form a group is anormal subgroup.28.17 The group of all non-singularn×nmatrices is known as the general linear
groupGL(n) and that with only real elements asGL(n,R). IfR∗denotes the
multiplicative group of non-zero real numbers, prove that the mapping Φ :
GL(n,R)→R∗, defined by Φ(M)=detM, is a homomorphism.
Show that the kernelKof Φ is a subgroup ofGL(n,R). Determine its cosets
and show that they themselves form a group.
28.18 The group of reflection–rotation symmetries of a square is known asD 4 ;let
Xbe one of its elements. Consider a mapping Φ :D 4 →S 4 , the permutation
group on four objects, defined by Φ(X) = the permutation induced byXon
the set{x, y, d, d′},wherexandyare the two principal axes, anddandd′
the two principal diagonals, of the square. For example, ifRis a rotation by
π/2, Φ(R) = (12)(34). Show thatD 4 is mapped onto a subgroup ofS 4 and, by
constructing the multiplication tables forD 4 and the subgroup, prove that the
mapping is a homomorphism.
28.19 Given that matrixMis a member of the multiplicative groupGL(3,R), determine,
for each of the following additional constraints onM(applied separately), whether
the subset satisfying the constraint is a subgroup ofGL(3,R):
(a) MT=M;
(b)MTM=I;
(c) |M|=1;
(d)Mij=0forj>iandMii=0.28.20 The elements of the quaternion group,Q,aretheset
{ 1 ,− 1 ,i,−i, j,−j, k,−k},withi^2 =j^2 =k^2 =−1,ij=kand its cyclic permutations, andji=−kand
its cyclic permutations. Find the proper subgroups ofQand the corresponding
cosets. Show that the subgroup of order 2 is a normal subgroup, but that the
other subgroups are not. Show thatQcannot be isomorphic to the group 4mm
(C 4 v) considered in exercise 28.11.28.21 Show thatD 4 , the group of symmetries of a square, has two isomorphic subgroups
of order 4. Show further that there exists a two-to-one homomorphism from the
quaternion groupQ, of exercise 28.20, onto one (and hence either) of these two
subgroups, and determine its kernel.
28.22 Show that the matrices
M(θ, x, y)=
cosθ −sinθx
sinθ cosθy
001
,
where 0≤θ< 2 π,−∞<x<∞,−∞<y<∞, form a group under matrix
multiplication.
Show that thoseM(θ, x, y)forwhichθ= 0 form a subgroup and identify its
cosets. Show that the cosets themselves form a group.28.23 Find (a) all the proper subgroups and (b) all the conjugacy classes of the symmetry
group of a regular pentagon.