Definitions and theorems Flashcards
What are the 4 things needed to be a group?
G1 ∀ a,b belonging to G, a*b belongs to G CLOSED UNDER BINARY OPERATION
G2 ∀ a,b,c in G, (ab)c=a(bc) ASSOCIATIVITY
G3 there is an element 1G in G st, ∀ a in G, 1a=a1=a IDENTITY
G4 ∀ g in G, there is an element g^-1 st gg^-1=g^-1g INVERSE
what is the subgroup criterion
h=/= empty set
∀ a,b, ab is in H
∀ a in H, a^-1 is in H
2nd and 3rd can be combined st its sufficient to prove ab^-1 remains in H
what is lagrange theorem
if H ≤ G then |G|=[G:H] |H|. [G:H] is the index of H in G.
In particular, |H| divides |G|
what is centralizer of S in G
CG(S)={g in G|gx=xg for all x in S}
what is normalizer of S in G
NG(S)={g in G| S^g = S}
what is subgroup of G generated by S
={x1,….,xm | m in N, xi belongs to SUS^-}
what is a conjugate in G of S
S^g={g^-1 x g | x belongs to S}
H K | = ?
|H| . |K| / | H ∩ K |
what does it mean if G acts on Ω
αg is a unique element st
(A1) g1,g2 in G, α in Ω, then α(g1g2)=(αg1)g2
(A2) for all α in Ω, α1=α
G orbit is…………………….?
α^G={αg | g belongs to G}
Stabilizer is…………..?
Gα={g in G | αg=α}
Class equation of a finite group
|G| = Σni = Σ[G:CG(xi)] |G| = |Z(G)| + Σni |G| = |Z(G)| + Σ[G:CG(xi)]
Right coset of a in G is?
Ha={ha | h∈H}
[G:H], the index of H in G is defined as what?
The number of right cosets of H in G. T
GLn(F) = …
Set of nxn matrices A with entries in F, such that det(A)=/=0
GLn(F) | = …
(q^n - 1)(q^n - q)(q^n - q^2)…(q^n - q^n-1)
What is a transposition?
A cycle of length 2
Every permutation in Sn can be written as what?
A product of transpositions
σ is an even permutation if …
σ is an even permutation if σ can be written as a product of an even number of transpositions (same for odd)
If σ can be written as a product of r1 transpositions and also a product of r2 transpositions, then what can you say about r1 and r2
either both even or both odd
An = ….
An = {σ |σ is an even permutation in Sn}
A3 = ….
A3 = {(1), (123), (132)}
For n ≥ 2, An is a subgroup of …
An ≤ Sn
