Groups of Transformations
Example of a Subgroup
The following is an example of a Cayley table of the symmetries of a regular hexagon(Group D6).
Symmetries of a Regular Hexagon(D_6)
Subgroups of the Symmetries of a Hexagon
The following are all subgroups because they include the identity element, include each of the inverses, are closed under the operation, and in this case we assume associativity.
The following are all subgroups excluding the trivial subgroups for the Cayley table above:
{R0,S0}
{R0,S1}
{R0,S2}
{R0,S3}
{R0,S4}
{R0,S5}
{R0,R3,S0}
{R0,R3,S1}
{R0,R3,S2}
{R0,R3,S3}
{R0,R3,S4}
{R0,R3,S5}
{R0,R1,R2,R3,R4,R5}
Symmetries of a Regular Triangle
Proof the Symmetries of a Regular Triangle form a Group
To prove this is a group, we need to show that the group is closed under the operation, contains the identity element, includes all the inverses, and has associativity(we assume this).
- We can construct a similar table above which shows that composing any two symmetries in the group returns a different symmetry. This means that the group is closed under the operation.
- It is clear that the group contains the identity element(e) which is just the "nothing operation".
- Because the group is generally not too large, we can go through and show that each inverse of each element is in the group. The identity is its own inverse, each reflection is its own inverse, and a rotation by 120 is the inverse of a rotation by 240 and vice versa. Thus, we can conclude that each symmetry or element in the group has its inverse in the group.
- In this case and in all other symmetry groups, we assume associativity.
- After checking each of the boxes that it closed under the operation, contains the identity element, includes all the inverses, and has associativity(we assume this), we can conclude that the symmetries of a regular triangle form a group.
Symmetries of a Regular Triangle
Group: {e, r_l,r_m,r_n, }
Subgroups:
The following are all subgroups because it includes the identity element, includes each of the inverses, is closed under the operation, and in this case we assume associativity.
{e}
{e,r_l}
{e,r_m}
{e,r_n}
{e,}
{e, r_l,r_m,r_n, }
Example of a subgroup proof
To form a subgroup, it must meet a few requirements. These requirements include being closed under the operation, has the identity element, includes the inverses of each element and we assume associativity in this case.
In this proof we will prove that {e,} is a subgroup of {e, r_l,r_m,r_n, }(D3).
First we will prove it is closed under the operation. Composing the identity element(e) with itself or either of the rotation symmetries returns that same symmetry. Composing the rotations with one another return a rotation of 360 degrees(e), a rotation of 120 degrees, or a rotation of 240 degrees. Each of these symmetries are included in the subgroup. We can conclude it is closed under the operation.
Then we will show that the subgroup contains each of the inverses along with the identity element. The group has the identity element(e) and the inverse of this element is just itself. The inverses of the rotation of 120 degrees and by 240 degrees are just each other. We can conclude the subgroup contains the inverses and the identity element.
Because we assume associativity, this concludes the proof. We have shown that {e,} is a subgroup of {e, r_l,r_m,r_n, }(D3).