Mathematical Methods for Physics and Engineering - Matematica.NET

GROUP THEORYmathematical details, a rotation about axis i can be represented by the operatorR i (θ), and the two rotations are connected by a relationship of the formR j (θ) =φ −1ij R i (θ)φ ij ,in which φ ij is the member of the full continuous rotation group that takes axisi into axis j.28.8 Exercises28.1 For each of the following sets, determine whether they form a group under the operationindicated (where it is relevant you may assume that matrix multiplicationis associative):(a) the integers (mod 10) under addition;(b) the integers (mod 10) under multiplication;(c) the integers 1, 2, 3, 4, 5, 6 under multiplication (mod 7);(d) the integers 1, 2, 3, 4, 5 under multiplication (mod 6);(e)all matrices of the form(a a− b0 bwhere a and b are integers (mod 5) and a ≠0≠ b, under matrix multiplication;(f) those elements of the set in (e) that are of order 1 or 2 (taken together);(g) all matrices of the form⎛⎝ 1 a 0 1 00⎞⎠ ,b c 1where a, b, c are integers, under matrix multiplication.28.2 Which of the following relationships between X and Y are equivalence relations?Give a proof of your conclusions in each case:(a) X and Y are integers and X − Y is odd;(b) X and Y are integers and X − Y is even;(c) X and Y are people and have the same postcode;(d) X and Y are people and have a parent in common;(e) X and Y are people and have the same mother;(f) X and Y are n×n matrices satisfying Y = PXQ,whereP and Q are elementsof a group G of n × n matrices.28.3 Define a binary operation • on the set of real numbers byx • y = x + y + rxy,where r is a non-zero real number. Show that the operation • is associative.Prove that x • y = −r −1 if, and only if, x = −r −1 or y = −r −1 . Hence provethat the set of all real numbers excluding −r −1 forms a group under the operation•.1070),