Mathematical Methods for Physics and Engineering - Matematica.NET

28.8 EXERCISESSimilarly compute C 2 C 2 , C 1 C 2 and C 2 C 1 . Show that each product coset is equal toC 1 or to C 2 ,andthata2× 2 multiplication table can be formed, demonstratingthat C 1 and C 2 are themselves the elements of a group of order 2. A subgrouplike H whose cosets themselves form a group is a normal subgroup.28.17 The group of all non-singular n × n matrices is known as the general lineargroup GL(n) and that with only real elements as GL(n, R). If R ∗ denotes themultiplicative 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 kernel K of Φ is a subgroup of GL(n, R). Determine its cosetsand show that they themselves form a group.28.18 The group of reflection–rotation symmetries of a square is known as D 4 ;letX be one of its elements. Consider a mapping Φ : D 4 → S 4 , the permutationgroup on four objects, defined by Φ(X) = the permutation induced by X onthe set {x, y, d, d ′ }, where x and y are the two principal axes, and d and d ′the two principal diagonals, of the square. For example, if R is a rotation byπ/2, Φ(R) = (12)(34). Show that D 4 is mapped onto a subgroup of S 4 and, byconstructing the multiplication tables for D 4 and the subgroup, prove that themapping is a homomorphism.28.19 Given that matrix M is a member of the multiplicative group GL(3, R), determine,for each of the following additional constraints on M (applied separately), whetherthe subset satisfying the constraint is a subgroup of GL(3, R):(a) M T = M;(b) M T M = I;(c) |M| =1;(d) M ij =0forj>iand M ii ≠0.28.20 The elements of the quaternion group, Q, aretheset{1, −1,i,−i, j, −j,k,−k},with i 2 = j 2 = k 2 = −1, ij = k and its cyclic permutations, and ji = −k andits cyclic permutations. Find the proper subgroups of Q and the correspondingcosets. Show that the subgroup of order 2 is a normal subgroup, but that theother subgroups are not. Show that Q cannot be isomorphic to the group 4mm(C 4v ) considered in exercise 28.11.28.21 Show that D 4 , the group of symmetries of a square, has two isomorphic subgroupsof order 4. Show further that there exists a two-to-one homomorphism from thequaternion group Q, of exercise 28.20, onto one (and hence either) of these twosubgroups, and determine its kernel.28.22 Show that the matrices⎛M(θ, x, y) = ⎝cos θ − sin θ xsin θ cos θ y0 0 1where 0 ≤ θ