MATH 817 Notes
JD Nir
[email protected]
www.math.unl.edu/∼jnir2/817.html
September 23, 2015
Lattice Theorem [4th Isomorphism Theorem] G group N E G. Let π : G → G/N be the canonical
map. There is a bijection
1-1
{subgroups of G/N } ←→ {subgroups of G that contain N }
onto
S
L ≤ G/N , L
7→
π −1 (L) =
gH
π(H) = G/N
gH∈L
N EH≤G
←[
H
Moreover, this bijection
1 preserves containment
T
2 preserves ’s
3 preserves joins: Given L, L0 ,
hπ −1 L ∪ π −1 L0 i = π −1 hL ∪ L0 i
hπ(H) ∪ π(H 0 )i = πhH ∪ H 0 i
N ≤ H ≤ G ⇒ [G : H] = [G/N : H/N ]
L ≤ G/N ⇒ [G/N : L] = [G : π −1 (L)]
N ≤ H E H ⇒ H/N E G/N
5 preserves normality:
L E G/N ⇒ π −1 (L) E G
6 If N ≤ H ≤ H 0 ≤ G corresponds to
4 preserves indicies:
L ≤ L0 ≤ G/N
then if H E H 0 then L E L0 and
H 0/H
∼
= L0/L
Pf of just the bijection assertion:
If ϕ : G → K is any group homomorphism, then H ≤ G ⇒ ϕ(H) ≤ K and L ≤ K ⇒ ϕ−1 (L) ≤ G.
k
{g|ϕ(g)∈L}
So, both functions, L 7→ π −1 (L) and H 7→ π(H) ≤
π −1 (L) ⊇ π −1 (eG/N ) = N .)
H/N ,
are well defined. (Also, L ≤ G/N ⇒
Given H ≤ G, N ≤ H, I claim π −1 π(H) = H :⊇ is clear
⊆:
g ∈ π −1 π(H) ⇒ π(g) ∈ π(H)
⇒ g · N = h · N, some h ∈ H
⇒ h−1 g ∈ N ⊆ H
⇒ h−1 g = g 0 , h0 ∈ H
⇒ g = hh0 ∈ H
1
JD Nir
[email protected]
MATH 817
∀L ≤ G/N , π(π −1 (L)) = L, since π is onto.
N = Z(D12 ) = hr3 i
Ex G = D12 ,
G = hr, s | r6 , s2 , srsri
G/N
= hr, s | r6 , s2 , srsr, r3 i = hr, s | s2 , srsr, r3 i
∼
= D6
D6
2 E 33 3
hri
hsi
D12
2
D E
hsri sr2
hri
3 E EE E 2
2 2
3
E
3
3
3
hs, r3 i
E 2
E
E 2
hsr, r3 i
E
2
hsr3 , r3 i
= hs, r3 i
hr3 i
hei
hsr2 , r3 i
Def A group is simple if G is not {e} and the only normal subgroups of G are {e} and G.
Ex 1
Z/p,
p prime, is simple.
2 An , n ≥ 5, is simple. [We will prove]
3 A4 is not simple:
V := {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}
V ≤ A4 by brute force. (e.g. (1 2)(3 4) · (1 3)(2 4) = (1 4)(3 2)
V E A4 :
HW: ∀σ ∈ Sn σ · (a1 a2 · · · am )σ −1 = (σ(a1 ) σ(s2 ) · · · σ(am ))
HW
∴ ∀σ ∈ S4 σ(1 2)(3 4)σ −1 = σ(1 2)σ −1 σ(3 4)σ −1 = (σ(1) σ(2))(σ(3) σ(4)) ∈ V
↑
σ is 1-1
∴ V E S4 (⇒ V E A4 )
Def A finite composition series of a group G is a chain
{e} = N0 ≤ N1 ≤ N2 ≤ · · · ≤ N` = G such that Ni−1 E Ni ∀i = 1, . . . , ` and
2
Ni/Ni−1
is simple.