28.8 EXERCISES
28.4 Prove that the relationshipX∼Y, defined byX∼YifYcan be expressed in
the form
Y=aX+b
cX+d,
witha,b,canddas integers, is an equivalence relation on the set of real numbers. Identify the class that contains the real number 1.
28.5 The following is a ‘proof’ that reflexivity is an unnecessary axiom for an equiva-
lence relation.
Because of symmetryX∼YimpliesY∼X. Then by transitivityX∼Yand
Y∼XimplyX∼X. Thus symmetry and transitivity imply reflexivity, which
therefore need not be separately required.
Demonstrate the flaw in this proof using the set consisting of all real numbers plus
the numberi. Show by investigating the following specific cases that, whether or
not reflexivity actually holds, it cannot be deduced from symmetry and transitivity
alone.(a) X∼YifX+Yis real.
(b)X∼YifXYis real.28.6 Prove that the setMof matrices
A=(
ab
0 c)
,
wherea, b, care integers (mod 5) anda=0=c, form a non-Abelian group
under matrix multiplication.
Show that the subset containing elements ofMthat are of order 1 or 2 do
not form a proper subgroup ofM,(a) using Lagrange’s theorem,
(b) by direct demonstration that the set is not closed.28.7 Sis the set of all 2×2 matrices of the form
A=(
wx
yz)
, wherewz−xy=1.Show thatSis a group under matrix multiplication. Which element(s) have order
2? Prove that an elementAhas order 3 ifw+z+1=0.
28.8 Show that, under matrix multiplication, matrices of the form
M(a 0 ,a)=(
a 0 +a 1 i −a 2 +a 3 i
a 2 +a 3 ia 0 −a 1 i)
,
wherea 0 and the components of column matrixa=(a 1 a 2 a 3 )Tare real
numbers satisfyinga^20 +|a|^2 = 1, constitute a group. Deduce that, under the
transformationz→Mz,wherezis any column matrix,|z|^2 is invariant.
28.9 IfAis a group in which every element other than the identity,I,hasorder2,
prove thatAis Abelian. Hence show that ifXandYare distinct elements ofA,
neither being equal to the identity, then the set{I, X,Y , XY}forms a subgroup
ofA.
Deduce that ifBis a group of order 2p,withpa prime greater than 2, thenB
must contain an element of orderp.
28.10 The group of rotations (excluding reflections and inversions) in three dimensions
that take a cube into itself is known as the group 432 (orOin the usual chemical
notation). Show by each of the following methods that this group has 24 elements.