Introduction to the Social Dimensions of Education : Consensus and Conflict T...
Symmetries of a square
1.
2. INTRODUCTION
A group (G,∗) is a set G together with a
law of composition (a,b) → 𝑎 ∗ 𝑏 that satisfies
the associative law, identity law, and inverses
law. The group (G,∗) is abelian if ∗ satisfy the
commutative law. A symmetry of a geometric
figure is a rearrangement of the figure
preserving the arrangement of its sides and
vertices as well as its distances and angles.
3. The aim of this investigation is to describe
the symmetries of a square and prove that the
set of the symmetries is a group. The symmetry
group of the square is denoted by D4 (dihedral
group). If D4 is a group, then we will determine
whether the group is abelian or nonabelian and
whether the group is cyclic or not. If it is cyclic,
then we will look for its generator. The
subgroups of D4 will also be examine in this
investigation. To solve this, I will do the
following:
4. 1. Describe the symmetries of a square
2. Give a Cayley’s Table for the
symmetries
3. Determine if D4 is closed under
operation ∗
4. List all the subgroups of D4
5. Classify each subgroups if it is cyclic
or non-cyclic.
5. MATHEMATICAL WORKING
Consider the square with vertices denoted by 1,
2, 3 and 4. We shall be concerned with all rigid motions
(rotations and reflections) such that the square will
look the same after the motion as before.
1
4 3
2
10. To make the table
Example: r1 ∗ d1 = m1
Let’s start with the position of r1 instead of
r0 . Then, flip it just like how you did to
obtain d1 .
= m1
1
2 3
4
12
3 4
27. SUBGROUPS
A subset H of a group G is a group if:
It’s closed under the operation
Its identity is the identity of G
Each element of the subset has an
inverse that’s also in the subset.
28. LAGRANGE THEOREM
The order of each subgroup of a finite
group G is a divisor of the order G.
Order – is the number of element in a set.
D4 is order 8
The subgroups of D4 are of orders 1, 2, 4, 8.
29. LIST ALL THE SUBGROUPS OF D4
Order 1 - r0
Order 2 - {r0 , r2 }, {r0 , m1 }, {r0 , m2 }, {r0 , d1 },
{r0 , d2 }
Order 4 - {r0 , r1 , r2 , r3 } , {r0 , r2 , m1 , m2 } ,
{r0 , r2 , d1 , d2 }
112. CONCLUSION
The elements of the group of the
symmetries of a square are the rotations and
reflections. Call this set D4 :
r = rotation 360°
r1 = rotate 90°
r2 = rotate 180°
r3 = rotate 270°
m1= reflect around vertical axis
m2= reflect around horizontal axis
d1 = reflect around x=y diagonal
d2 = reflect around x=-y diagonal
113. CONCLUSION
Set D4 = {r0 , r1 , r2 , r3 , m1 , m2 , d1 , d2 }
together with a law of composition satisfies the
associative law, identity law, and inverses law. The
identity element of D4 is r0 and every element of set
D4 has unique inverses. Therefore, (D4 , ∗) is a
group. This group is nonabelian and also not a
cyclic group because it did not satisfy the
commutative law. Clearly, every cyclic group is
abelian.
114. CONCLUSION
D4 is of order 8 and its subgroups are of
order 1, 2 and 4. It has 9 subgroups. It has two
non-cyclic subgroups ({r0 , r2 , m1 , m2 } , {r0 , r2 ,
d1 , d2 }) and seven cyclic subgroups ( r0 , {r0 ,
r2 }, {r0 , m1 }, {r0 , m2 }, {r0 , d1 }, {r0 , d2 }, {r0 , r1
, r2 , r3 }. Even though D4 is not a cyclic group, it
does not mean that it has no cyclic subgroups.