Group Theory
1 Summary
Think of groups as a set of different operations.
Fundamental identity group = identity operation = Do Nothing.
Composing operations which End up Doing nothing is interesting.
2 Dihedral D6 aka S3
- D6 (Dihedral group): Symmetries of equilateral triangle {identity, two rotations, three reflections}
- S3 (3-symmetric group): A queue of 3 people in order A,B,C: {identity, pop A + add to back, pop C + add to front, swap A B, swap A C, swap B C }
Notice the operations of D6 and S3 are isomorphic!
Operations
operations are {identity, rotate 120, rotate -120, reflect 1st vertex, reflect 2nd vertex, reflect 3rd vertex}
3 C1 S1 S0
- Group with 1 identity operation
- Analogous to the action of doing nothing or Identity operation
| \(\times\) | 1 |
|---|---|
| 1 | 1 |
Operations
operations are {1: identity}
4 C2
- group of order 2
| \(\times\) | 1 | a |
|---|---|---|
| 1 | 1 | a |
| a | a | 1 |
Operations
operations are {1: identity, a: swap}
- Given a 2-char string, the action of swapping 2 characters defines a symmetry group.
5 C3
| \(\times\) | 1 | a | b |
|---|---|---|---|
| 1 | 1 | a | b |
| a | a | b | 1 |
| b | b | 1 | a |
Operations
operations are {1: identity, a: rotate 120, b: rotate -120}
rotations of a triangle
- a*a = b, rotate 240 which is basically rotate -120
We also notice on a META level we can swap the operations ‘a’ and ‘b’ and it would still remain a group.
This symmetry of swapping the ‘a: rotate 120’ and ‘b: rotate -120’ tells us at Meta level this group itself has a symmetry group C2.
- This Meta level symmetry on a group is called an AUTOMORPHISM
- To find an AUTOMORPHISM check if you can rearrange the non-identity elements in a pre-existing symmetry pattern Cn and still form a group
- Permutation of {a,b} is \(_2 P _2=2\)
\[ Aut(C_3)=C_2\]
What about C2, what is it’s automorphism?
Since C2 only has 1 non-identity element, it can only do nothing (C1)
\[ Aut(C_2) = C_1\] \[ Aut(C_1) = C_1\]
6 C4
| \(\times\) | 1 | a | a2 | a3 |
|---|---|---|---|---|
| 1 | 1 | a | a2 | a3 |
| a | a | a2 | a3 | 1 |
| a2 | a2 | a3 | 1 | a |
| a3 | a3 | 1 | a | a2 |
Operations
operations are : {1: identity, a: rotate 90, a2: rotate 180, a3: rotate 270}
rotations of a square
7 4 distinct elements can build V4 and C4
- V4 notice any element squared is an identity
| \(\times\) | 1 | a | b | c |
|---|---|---|---|---|
| 1 | 1 | a | b | c |
| a | a | 1 | c | b |
| b | b | c | 1 | a |
| c | c | b | a | 1 |
V4 Operations
Operations is {1: identity, a: reflect_horizontal, b: reflect_vertical, c: rotate 180}
V4 can represent symmetry of a rectangle:
What is it’s automorphism?
There are \(_{3} P_{3}=6\) ways to rearrange {a,b,c} and each of them still result in groups meaning
\[Aut(V_4) = D_6\]
WARN: You must check if all permutations of group operations still satify the group properties.(We omit it here for brevity)
7.1 Disguised C4
Let’s make another group
we can define b * b = a, which implies b * c = c * b=1 which implies c * c = a
This group is actually just C4 which we already did in the previous section!
| \(\times\) | 1 | a | b | c |
|---|---|---|---|---|
| 1 | 1 | a | b | c |
| a | a | 1 | c | b |
| b | b | c | a | 1 |
| c | c | b | 1 | a |
What is it’s automorphism?
Remember C4 is the rotational symmetry of a square {identity, rotate 90, rotate -90, rotate 180}
We can only swap {rotate 90, rotate -90} and REMEMBER the act of swapping is C2 meaning
\[Aut(C_4)=C_2\]
8 Revisit C4 and learn about Generators
| \(\times\) | 1 | a | a2 | a3 |
|---|---|---|---|---|
| 1 | 1 | a | a2 | a3 |
| a | a | a2 | a3 | 1 |
| a2 | a2 | a3 | 1 | a |
| a3 | a3 | 1 | a | a2 |
- Find the Generated(a2) aka generated set of a2 ?
- Keep multiplying an element by itself
- HINT: Look at diagonal to see which element may have interesting generated sets.
- Solution: {1,a2}
- {1,a2} is called the generated SUBGROUP C2
- Order(a2,C4)=2 ,Order of an element wrt to a group is cardinality of the generated set of the element
- Keep multiplying an element by itself
8.1 Lagrange Theorem
- Order is an overloaded term.
- Order of a group = number of operations/elements in the group. ex: Order(C4)
- Order of an element wrt to a group = Order of a subgroup. ex: Order(a2,C4)
Lagrange Theorem: Order of an element wrt to a group Always DIVIDES Order of the group.
Example: Order(a2,C4)=2 divides Order(C4)=4
8.1.1 Deduce Operations of groups w/ Lagrange
- Given Group with Order(C6)
- then we know there must be a subgroup with operations of order 3, since 3 divides 6.
- order 3 subgroup has to have operations {1, a, a2}
8.2 Deduce Prime group property w/ Lagrange
Prime order groups can only have a subgroup of itself and the identity group C1
Since a prime number can only be divided by itself and 1.
9 Direct products C6 = C3 * C2
| \(\times\) | 1 | a | a2 | b | ab | a2 b |
|---|---|---|---|---|---|---|
| 1 | 1 | a | a2 | b | ab | a2b |
| a | a | a2 | a3 | ab | a2b | b |
| a2 | a2 | a3 | 1 | a2b | b | ab |
| a3 | a3 | 1 | a | 1 | a | a2 |
| ab | ab | a2b | b | a | a2 | 1 |
| a2b | a2b | b | ab | a2 | 1 | a |
