Page 105 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 105
mož Moravec: Some Topics in the Theory of Finite Groups 93
We will briefly tackle the problem of classifying extensions of abelian groups. It will
be shown that these are, up to equivalence, in 1-1 correspondence with the elements of
a certain second cohomology group. Cohomological group theory is an area on its own,
and we will not go deeply into it. We refer to [3] and [8] for further details.
3.4.1 Basic notions
A group extension of a group N by a group G is a short exact sequence
1 /N µ /E ε /G /1.
From the above it clearly follows that µ is injective, ε is surjective, M = im µ = ker ε is a
normal subgroup of E , M =∼ N , and E /M =∼ G .
A morphism between extensions N / µ / E ε / / G / 1 and N / µ / E ε / / G
is a triple of group homomorphisms (α, β , γ) such that the following diagram commutes:
N / µ /E ε / /G .
α βγ
µ ε
N/ /E / /G
The collection of all group extensions and morphisms between them is a category. A
morphism of the type
N / µ /E ε / /G
1 β1
/ µ ε / /G
N /E
is said to be an equivalence of extensions.
3.4.2 Semidirect products
Suppose that H and N are groups and that we have a homomorphism α: H → Aut(N ).
The (external) semidirect product H α N of N and H is the set of all pairs (h, n ), where
h ∈ H , n ∈ N , with the operation
(h 1 , n 1 )(h 2 , n 2 ) = (h 1 h 2, n h α n 2 ).
1 2
This is a group with the identity element (1H , 1N ), and the inverse of (h, n ) is (h−1, n −(hα)−1 ).
We have embeddings H → H α N and N → H α N given by h → (h, 1N ) and n →
(1H , n ), respectively. If H ∗ and N ∗ are images of these maps, then N ∗ H α N , H ∗ ∩N ∗ = 1
and H ∗N ∗ = H α N . We say that H α N is the internal semidirect product of N ∗ and H ∗.
We will briefly tackle the problem of classifying extensions of abelian groups. It will
be shown that these are, up to equivalence, in 1-1 correspondence with the elements of
a certain second cohomology group. Cohomological group theory is an area on its own,
and we will not go deeply into it. We refer to [3] and [8] for further details.
3.4.1 Basic notions
A group extension of a group N by a group G is a short exact sequence
1 /N µ /E ε /G /1.
From the above it clearly follows that µ is injective, ε is surjective, M = im µ = ker ε is a
normal subgroup of E , M =∼ N , and E /M =∼ G .
A morphism between extensions N / µ / E ε / / G / 1 and N / µ / E ε / / G
is a triple of group homomorphisms (α, β , γ) such that the following diagram commutes:
N / µ /E ε / /G .
α βγ
µ ε
N/ /E / /G
The collection of all group extensions and morphisms between them is a category. A
morphism of the type
N / µ /E ε / /G
1 β1
/ µ ε / /G
N /E
is said to be an equivalence of extensions.
3.4.2 Semidirect products
Suppose that H and N are groups and that we have a homomorphism α: H → Aut(N ).
The (external) semidirect product H α N of N and H is the set of all pairs (h, n ), where
h ∈ H , n ∈ N , with the operation
(h 1 , n 1 )(h 2 , n 2 ) = (h 1 h 2, n h α n 2 ).
1 2
This is a group with the identity element (1H , 1N ), and the inverse of (h, n ) is (h−1, n −(hα)−1 ).
We have embeddings H → H α N and N → H α N given by h → (h, 1N ) and n →
(1H , n ), respectively. If H ∗ and N ∗ are images of these maps, then N ∗ H α N , H ∗ ∩N ∗ = 1
and H ∗N ∗ = H α N . We say that H α N is the internal semidirect product of N ∗ and H ∗.
