MATH 817 Notes
JD Nir
[email protected]
www.math.unl.edu/∼jnir2/817.html
November 4, 2015
Recall:
If H and K are groups and ρ : K → Aut(H) is a group homomorphism, then H oρ K denotes the
set H × K equipped with the operation
mult. in H
↓
mult. in K
↓
(h1 , k1 ) · (h2 , k2 ) = (h1 · hk21 , k1 · k2 )
where hk21 := ρ(k1 )(h2 ).
Theorem 1 Given H, K, ρ, H oρ K is a group.
2 If G is a group, H E G, K ≤ G such that H ∩ K = {e} and HK = G then
H oρ K ∼
=G
(h,k)7→h·k
where ρ is given by the action of K on H via conjugation. (i.e., hk := khk −1 )
Note 1 If ρ is trivial, H oρ K = H × K
| {z }
product of groups
2 If 2 of the Theorem, if K E G too, then K acts trivially on H and we get H × K ∼
= G.
Ex D2n = G H = hri E G
K = hsi ≤ G
H ∩ K = {e} HK = G
∼ H oρ K.
∴ G=
∼
=Z/2
where ρ : K −→ Aut(H) ∼
= (Z/n)×
ρ(s) = (r 7→ r−1 )
ρ(s)(r) = r−1
rs = r−1
Why? srs−1 = srs = r−1
So, D2n ∼
= Z/n ×ρ Z/2, ρ : Z/2 → Aut(Z/n) = (Z/n)×
17→−1
Pf 1
• Associative axiom: exercise
• eHoρ K = (eH , eK )
(eH , eK ) · (h, k) = (eH · heK , eK · k) = (h, k)
(h, k) · (e, e) = (h · ekH , k · eK ) = (h, k)
1
MATH 817
JD Nir
[email protected]
−1
• Claim: (h, k)−1 = ((h−1 )k , k −1 )
−1
−1
−1
−1
((h−1 )k , k −1 ) · (h, k) = ((h−1 )k · hk , k −1 k) = ((h−1 h)k , e) = (e, e)
−1
−1
(h, k)((h−1 )k , k −1 ) = (h · ((h−1 )k )k , e) = (h · (h−1 )e , e) = (e, e)
2 follows the from the calculations done on Monday.
Ex Say G is a group of order 30. Recall:
• ∃H E G, H = hxi, |x| = 15
• ∃K ≤ G, k = hyi, |y| = 2
• G∼
= hx, y | x15 , y 2 , yxy −1 x−j i with j ∈ {1, 4, 11, 14}
We didn’t show that all four possibilities occur.
Let H = hx | x1 5i ∼
= Z/15
K = hy | y 2 i ∼
= Z/2
Aut(H) ∼
= (Z/15)× has 4 elements of order 1 or 2: 1, 4, 11, 14 ∈ (Z/15)× and these correspond to
α1 , α2 , α3 , α4 ∈ Aut(H), α1 (x) = x, α2 (x) = x4 , α3 (x) = x11 , α4 (x) = x14 = x−1
Define a group homomorphism ρi : K → Aut(H) bu ρi (y) = αi , i = 1, 2, 3, 4.
(Such ρi ’s exist by the Universal Mapping Property of presentations)
This gives me 4 groups
H oρi K i = 1, 2, 3, 4
Also, H oρi K = hx, y | x15 , y 2 , yxy −1 x−j i
j = 1 if i = 1
j = 4 if i = 2
j = 11 if i = 3
j = 14 if i = 4
Each has order 30.
In fact, thse four groups are not isomorphic to each other
H = hy | y 2 i ∼
= Z/2
H = hx | ∅i ∼
=Z
Aut(H) = {±1}
ρ : K → Aut(H)
ρ(y) = (x 7→ x−1 )
ρ(y)(x) = x−1
H oρ K = hx, y | y 2 , yxyxi = D∞
#G = 75 = 52 · 3
n5 = 1
∴ ∃H E G, #H = 25
∃K ≤ G, #K = 3
2
MATH 817
H ∩ K = {e}
#HK = 75
∴ HK = G.
∴ G∼
= H oρ K, for some ρ : K → Aut(H).
Ex ∃ at least two non-abelian groups of order p3 , p = prime.
1 H = Z/p2 K = Z/p
# Aut(G) = p2 − p = p(p − 1)
∴ ∃α ∈ Aut(H), |α| = p
⇒ ∃ρ : K → Aut(H), ρ(1) = α
2
H oρ K = hx, y | xp , y, yxy −1 = xj i
j 6= 1, 1 < j < p2 , j 2 ≡ 1(p2 )
2 H = Z/p × Z/p
Aut(H) ∼
= GL2 (Z/p)
# Aut(H) = (p2 − 1)(p2 − p)
∴ p | # Aut(H)
⇒ ∃α ∈ Aut(H), |α| = p
1 1
α↔
∈ GL2 (Z/p)
0 1
...
hx, y, z | xp , y p , z p , xyx−1 x−1 , xzx−1 , z −1 , zyz −1 = xyi
3
JD Nir
[email protected]