Chapter 4 CONJUGACY Flashcards
Conjugate of S
S^g={g^-1 x g | x ∈ S}
Conjugacy Class of S
x^g={g^-1 x g | g ∈ G}
|S| = …
|S| = |S^g|
If S ≤ G, then …
S^g ≤ G
if x and y are conjugate in G then x and y…
have the same order
1^G = …
{1}
Suppose G a group, then G is …
G is the disjoint union of its conjugacy classes.
Proof: x~y <=> x^g=y
prove that ~ is an equivalence relation. REFLEXIVE, SYMMETRIC, TRANSITIVE
let G group, N = NG(S)
let {gi | i ∈ I} be a complete set of right coset representatives for N in G.
conjugates of S are {S^gi | i ∈ I} and S^gi = S^gj <=> …
gi = gj.
If further, G is finite, number of conjugates of S is [G:N] = |G|/|N|
start of proof: S^g is a typical conjugate. g=ngi for some gj.
then S^g = S^ngj = (S^n)gj = S^gj
NG({x}) = …
CG({x}).
[x^g= x becomes g-1 x g =x which is the centralizer]
let gi be a complete set of right coset representatives for C in G. then x^gi =x^gj <=> …
gi = gj
Furthermore if G is finite, then |x^G| = [G: CG(x)] and |x^G| | |G|
S≤ NG(S)≤ G and so by lagrange’s theorem what can we say NG(S) is equal to
either S=NG(S) or G=NG(S)
Class equation for finite groups
let x1…xk be representatives from each of the k conjugacy classes of G. set ni = |xi ^ G|
assume n1=n2=….=nl =1 , nj>1 for j>l
|G| = (1 to n)Σni = (1 to n)Σ[G:CG(xi)] |G| = |Z(G)| + (l+1 to k)Σni |G| = |Z(G)| + (l+1 to k)Σ[G:CG(xi)]
if p is prime, what is a p group?
|G| = p^r with r∈NU{0}
example: |G| = 125= 5^3. G is a 5-group
If G is a p group, what can be said about Z(G)
Z(G) =/= 1.
proof using class equation of finite groups
if f:a->b then g-1 f g = …
(a,b belong to omega = {1,….,n}
ag->bg