APP. B] ALGEBRAIC SYSTEMS 455
Fig. B-8
(a) AsymmetryσofSis a rigid one-to-one correspondence betweenSand itself. (Here rigid means that distances
between points do not change.) The groupGof symmetries ofSis the set of all symmetries ofSunder composition
of mappings.
(b) There are eight symmetries as follows. Forα = 0 ◦,90◦, 180◦, 270◦, letσ(α)be the symmetry obtained by
rotatingSabout its centerαdegrees, and letτ(α)be the symmetry obtained by reflectingSabout the y-axis and
then rotatingSabout its centerαdegrees. Note that any symmetryσofSis completely determined by its effect
on the vertices ofSand henceσcan be represented as a permutation inS 4. Thus:
σ( 0 ◦)=
(
1234
1234
)
,σ( 90 ◦) =
(
1234
2341
)
,
σ( 180 ◦)=
(
1234
3414
)
,σ( 270 ◦)=
(
1234
4123
)
τ( 0 ◦)=
(
1234
2143
)
,τ( 90 ◦) =
(
1234
3214
)
,
τ( 180 ◦)=
(
1234
4321
)
,τ( 270 ◦)=
(
1234
1432
)
(c) Leta=σ( 90 ◦)andb=τ( 0 ◦). Thenaandbform a maximum set of generators ofG. Specifically,
σ( 0 ◦)=a^4 ,σ( 90 ◦)=a, σ( 180 ◦)=a^2 ,σ( 270 ◦)=a^3
τ( 0 ◦)=b, τ ( 90 ◦)=ba, τ ( 180 ◦)=ba^2 ,τ( 270 ◦)=ba^3
andGis not cyclic so it is not generated by one element. (One can show that the relationsa^4 =e,b^2 =e, and
bab=a−^1 completely describeG.)
B.12.LetGbe a group and letAbe a nonempty set.
(a) Define the meaning of the statement “Gacts onA.”
(b) Define the stabilizerHaof an elementa∈A.
(c) Show thatHais a subgroup ofG.
(a) Let PERM(A) denote the group of all permutations ofA. Letψ:G→PERM(A) be any homomorphism. Then
Gis said to act onAwhere each elementginGdefines a permutationg:A→Aby
g(a)=(ψ(g))(a)
(Frequently, the permutationg:A→Ais given directly and hence the homomorphism is implicitly defined.)
(b) The stabilizerHaofa∈Aconsists of all elements ofGwhich “fixa,” that is,
Ha={g∈G|g(a)=a}