Groups Flashcards
(21 cards)
What is C_n?
The cyclic group, rotational symmetry for
n-gon with directed sides
What is D_n?
Dihedral group, rotation and reflection for n-gon with no direction on sides.
What is the order of C_n?
n
What is the order of D_n?
2n
What is the order of S_n?
n!
What is the order of Z_n?
n
What is S_n?
Permutations of n objects.
What is Z_n?
Integers under addition mod n
What is the condition for a group to be Abelian?
Composition law is commutative
How many binary operators act on a Field?
2, addition and multiplication.
How many binary operators act on a group?
1, multiplication
List the 6 field axioms.
- Closure
- Commutativity
- Associativity
- Distributivity
- Identities exist
- Inverses exist
List the 4 group axioms
- Closure
- Associativity
- Unique Identity
- Inverses
List all conjugacy classes of D3
(e) - trivial class
(c, c^2)
(m, mc, mc^2)
Define a subgroup of G
A subset of G which itself follows the composition law of G and obeys all group axioms.
Define the coset of g of G with a given subgroup H = {h1,h2,…,hn}
Coset = gH = {gh1, gh2, gh3,…,ghn}
What are the two options for the coverage of two coset of G?
Completely overlapping or completely disjoint
What are the three axioms of an equivalence relation ~?
- Reflexive a~a
- Symmetric a~b implies b~a
- Transitive a~b and b~c implies a~c
State the equivalence relation axioms for a chosen for example equal gender.
Reflexive: a~a
Mark is the same gender as Mark.
Symmetric: a~b => b~a
If Dave is the same gender as Mark, then Mark is the same gender as Dave.
Transitive: a~b & b~c => a~c
If Dave is the same gender as Mark, and Steve is the same gender as Dave, then Steve is the same gender as Mark.
Define the normal subgroup H of G.
gHg^1 = H for all g in G
Or gH = Hg is sufficient.
I.e all the conjugates of h are also contained in H
The product of two cosets g1H, g2H =? Where H is a normal subgroup of G
(g1H)(g2H) = g1g2H
Because H is normal subgroup:
g1Hg2H = g1HHg2 = g1Hg2 = g1g2H
