Group extensions

1. Split and non-split extensions: ⋊ and .
2. Conventions and tables
3. The action of Q on N
4. Example: C_2^n-1⋊C₂
5. Central and stem extensions

Split (⋊) and non-split (.) extensions

If N ◃ G is a normal subgroup with quotient Q = G/N, equivalently there is an exact sequence

1 → N → G → Q → 1,

then G is an extension of Q by N. (The terminology is somewhat unfortunate.) When the sequence splits, that is G has a subgroup isomorphic to Q that meets N trivially, G is called a split extension of Q by N, or a semi-direct product of N and Q, written

G = N⋊Q or G = N:Q

Otherwise G is a non-split extension of Q by N, written

G = N.Q

Note that the notation G = N.Q often stands for 'G is an extension which is not necessarily split' (e.g. in ATLAS), but it is only used for non-split extensions in the database.

Every finite group can be built up from simple groups using extensions, so these constructions are fundamental for describing general groups.

Conventions and tables

We exclude the trivial split extension G=N×Q from the list of extensions. For example:

S₃ = C₃⋊C₂	valid	C₃◃S₃ with C₂ quotient, and there is a copy C₂ < S₃ meeting C₃ trivially
C₄ = C₂.C₂	valid	C₂◃C₄ with C₂ quotient, and there is no other C₂ < C₄ meeting C₂ trivially
C₆ = C₃⋊C₂	no	C₆≅C₃×C₂, and direct products are excluded from split extensions
S₃ = C₃.C₂	no	S₃ ≅ C₃⋊C₂, and semidirect products are excluded from non-split extensions

There may be many different groups that are extensions of a given group Q by a given group N, in which case they are ordered, and the index is shown in the tables as a subscript of ⋊ or .

C₂.₁C₂² = D₄	1^st non-split extension of C₂² by C₂
C₂.₂C₂² = Q₈	2^nd non-split extension of C₂² by C₂

Most groups in the database are extension N⋊Q or N.Q for various choices of N and Q. For each choice, G appears in the table Name(Q)byName(N).html; for instance, C2^2byC2.html for the extensions of C₂² by C₂ as above.

The action of Q on N

When G=N⋊Q is a split extension, a copy of Q in G acts on N by conjugation, which gives a homomorphism Q→Aut N. Conversely, given N, Q and φ: Q→Aut N,

there is a unique split extension of Q by N with action φ, the semidirect product N⋊_φQ = {(n,q) | n∈N, q∈Q} with group operation (n,q)(n',q')=(n·φ(q)(n'),q·q')
the non-split extensions of Q by N with action φ are classified by the non-trivial elements of the cohomology set H²(Q,N). (When N is abelian, this is an abelian group.) Note that some classes may give isomorphic G's, as abstract groups.

When G=N.Q is a non-split extension with N abelian, the action of G on N by conjugation descends to φ: Q=G/N→Aut N. When N is nonabelian, G→Aut N need not descend to Q=G/N→Aut N, and having such a homomorphism φ is neither a necessary nor a sufficient condition to having the corresponding extension. Dividing out by inner automorphisms does give a well-defined homomorphism G→Aut N/Inn N=Out N (outer automorphism group), though it is not a particularly good invariant to classify extensions.

Example: C_2^n-1⋊C₂

n=2. The only homomorphism φ: C₂→Aut(C₂)=C₁ is trivial, the group H²(C₂,C₂)=C₂ has only one non-trivial class, and

C₂⋊C₂ = C₂²	unique split extension with trivial action(direct product)
C₂.C₂ = C₄	unique non-split extension with trivial action.

n=3. There are two homomorphisms C₂→Aut(C₄)=C₂. The trivial one gives

C₄⋊C₂ = C₂×C₄	unique split extension (direct product)
C₄.C₂ = C₈	unique non-split extension

and the non-trivial one

C₄⋊C₂ = D₄	unique split extension with non-trivial φ
C₄.C₂ = Q₈	unique non-split extension with non-trivial φ

n>3. There are four homomorphisms

φ: C₂ → Aut(C_2^n-1)= (ℤ/2^n-1ℤ)^× (≅C_2^n-3×C₂), sending the non-trivial generator g of C₂ to 1, -1, 2^n-3-1 and 2^n-3+1. The corresponding split extensions are

φ: g ↦ 1	C_2^n-1⋊_φC₂ = C_2^n-1×C₂	direct product
φ: g ↦ -1	C_2^n-1⋊_φC₂ = C_2^n-1⋊₁C₂ = D_2^n-1	dihedral group
φ: g ↦ 2^n-3-1	C_2^n-1⋊_φC₂ = C_2^n-1⋊₂C₂ = SD_2ⁿ	semidihedral group
φ: g ↦ 2^n-3+1	C_2^n-1⋊_φC₂ = C_2^n-1⋊₃C₂ = M_n(2)	modular maximal-cyclic group

In the two latter cases, H²(C₂,C_2^n-1)=0, so there are no non-split extensions with such actions. In the first two, H²(C₂,C_2^n-1)=C₂ and there is exactly one:

φ: g ↦1	C_2^n-1._φC₂ = C_2ⁿ	cyclic group
φ: g ↦-1	C_2^n-1._φC₂ = Q_2ⁿ	(generalized) quaternion group

Central and stem extensions

A non-split extenson G=N.Q is a central extension if N is contained in the center of G (and is, in particular, abelian). It is a central stem extension if, in addition, N is contained in the derived subgroup G'. For perfect groups G, their central stem extensions are classified by the theory of Schur covers.

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