SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

THE DIHEDRAL GROUP OF THE SQUARE

Consider a cardboard square as shown in Figure 1. There are eight motions of this square which, when performed one after the other, form a group called the Dihedral Group of the Square”. They are:

I – 0^{0} rotation (clockwise, about center O, in plane of cardboard)

R – 90^{0 }rotation (clockwise, about center O, in plane of cardboard)

R_{1} – 180^{0} rotation (clockwise, about center O, in plane of cardboard)

R_{2} – 270^{0} rotation (clockwise, about center O, in plane of cardboard)

H – reflection about horizontal axis AB (180^{0} flip through space)

V – reflection about vertical axis EF (180^{0} flip through space)

D – reflection about diagonal 1-O-3 (180^{0} flip through space)

D_{1} – reflection about diagonal 2-O-4 (180^{0} flip through space)

The Dihedral Group of the Square then is given by G = [ I, R, R_{1}, R_{2}, H, V, D, D_{1} ].
Multiplication in G consists of
performing two of these motions
in succession. Thus the product
HR corresponds to first
performing operation H, then
operation R. A multiplication
table for G is shown in Figure 2.
Entries in the table contain the
product XY where X corresponds
to the row and Y corresponds to
the column. Thus in the table HR
= D_{1}.

The eight motions I, R, R_{1}, R_{2}, H,
V, D, D_{1} can be represented as
permutations of the numbers 1, 2,
3, and 4 where these numbers
correspond to the four corners of
the square as shown in Figure 1.
Figure 3 shows their permutation
representation. Thus G can also be represented as the set

G = [ (1), (1432), (13)(42), (1234), (14)(23), (12)(43), (42), (13) ]

of permutations of the vertices. In this case multiplication corresponds to the multiplication of permutations.

I |
(1) |
0 |

R |
(1432) |
90 |

R |
(13)(42) |
180 |

R |
(1234) |
270 |

H |
(14)(23) |
reflection about horizontal axis AB |

V |
(12)(43) |
reflection about vertical axis EF |

D |
(42) |
reflection about diagonal 1-O-3 |

D |
(13) |
reflection about diagonal 2-O-4 |

Figure 3

Subgroups The subgroups of G are shown in Figure 4 along with their relationship to one another.

_________________________________________________________

_________________________________________________________

Group G is not cyclic. It is generated by the two elements R and H. From the multiplication table it can be seen that

R^{0} = I R = R R^{2} = R_{1} R^{3} = R_{2}

H^{0} = I H = H HR = D_{1} HR^{2} = V HR^{3} = D

Thus we see that the elements of G can be represented uniquely as H^{i}R^{j}_{ }with i = 0,1 and

j = 0, 1, 2, 3 i.e.

H^{0}R^{0} = I H^{0}R^{1} = R H^{0}R^{2} = R_{1} H^{0}R^{3} = R_{2}

H^{1}R^{0} = H H^{1}R^{1} = D_{1} H^{1}R^{2} = V H^{1}R^{3} = D

Transforms. Figure 5 shows the transforms of each element a of G for each value of x. If we read across on the rows we see the conjugates of each element a. Thus the rows represent the different conjugate classes into which the group is partitioned. In some cases different rows give the same conjugate class. We can list the conjugate classes:

class 1 = { I }

class 2 = { R, R_{2} }

class 3 = { R_{1} }

class 4 = { H, V }

class 5 = { D, D_{1} }

These transforms correspond to automorphic mappings of the elements. Thus if we wish to
know what subgroup a subgroup is mapped into by a particular automorphic mapping we can
read it off from the table. For example, the subgroup [ I, H, V, R_{1} ] is mapped into [ I, H, V, R_{1} ]
by the automorphic mapping T_{R}(a) : a
x^{-1}ax i.e. it is mapped into itself.

References

Birkhoff, Mac Lane. A Survey of Modern Algebra. Chap. VI

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-control, self-restraint, self-discipline basic to so much in life

[ Home ] [ Up ] [ Info ] [ Mail ]