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
If N ◃ G is a normal subgroup with quotient Q = G/N, equivalently there is an exact sequence
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
Otherwise G is a non-split extension of Q by N, written
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.
|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.
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.
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|
|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: g ↦ 1||C2n-1⋊φC2 = C2n-1×C2||direct product|
|φ: g ↦ -1||C2n-1⋊φC2 = C2n-1⋊1C2 = D2n-1||dihedral group|
|φ: g ↦ 2n-3-1||C2n-1⋊φC2 = C2n-1⋊2C2 = SD2n||semidihedral group|
|φ: g ↦ 2n-3+1||C2n-1⋊φC2 = C2n-1⋊3C2 = Mn(2)||modular maximal-cyclic group|
|φ: g ↦1||C2n-1.φC2 = C2n||cyclic group|
|φ: g ↦-1||C2n-1.φC2 = Q2n||(generalized) quaternion group|
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.