Group extensions

1. Split and non-split extensions: ⋊ and .
2. Conventions and tables
3. The action of Q on N
4. Example: C2n-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

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

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: C2n-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

C2C2 = 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

C4C2 = C2×C4 unique split extension (direct product)
C4.C2 = C8 unique non-split extension
and the non-trivial one
C4C2 = 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(C2n-1)= (ℤ/2n-1ℤ)× (≅C2n-3×C2),
sending the non-trivial generator g of C2 to 1, -1, 2n-3-1 and 2n-3+1. The corresponding split extensions are
φ: g ↦ 1 C2n-1φC2 = C2n-1×C2 direct product
φ: g ↦ -1 C2n-1φC2 = C2n-11C2 = D2n-1 dihedral group
φ: g ↦ 2n-3-1 C2n-1φC2 = C2n-12C2 = SD2n semi​dihedral group
φ: g ↦ 2n-3+1 C2n-1φC2 = C2n-13C2 = Mn(2) modular maximal-cyclic group
In the two latter cases, H2(C2,C2n-1)=0, so there are no non-split extensions with such actions. In the first two, H2(C2,C2n-1)=C2 and there is exactly one:
φ: g ↦1 C2n-1.φC2 = C2n cyclic group
φ: g ↦-1 C2n-1.φC2 = Q2n (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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