1. Split and non-split extensions: ⋊ and .
2. Conventions and tables
3. The action of Q on N
4. Example: C_{2n-1}⋊C_{2}
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.
S_{3} = C_{3}⋊C_{2} | valid | C_{3}◃S_{3} with C_{2} quotient, and there is a copy C_{2} < S_{3} meeting C_{3} trivially |
C_{4} = C_{2}.C_{2} | valid | C_{2}◃C_{4} with C_{2} quotient, and there is no other C_{2} < C_{4} meeting C_{2} trivially |
C_{6} = C_{3}⋊C_{2} | no | C_{6}≅C_{3}×C_{2}, and direct products are excluded from split extensions |
S_{3} = C_{3}.C_{2} | no | S_{3} ≅ C_{3}⋊C_{2}, 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 .
C_{2}._{1}C_{2}^{2} = D_{4} | 1^{st} non-split extension of C_{2}^{2} by C_{2} |
C_{2}._{2}C_{2}^{2} = Q_{8} | 2^{nd} non-split extension of C_{2}^{2} by C_{2} |
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 C_{2}^{2} by C_{2} 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: C_{2}→Aut(C_{2})=C_{1} is trivial, the group H^{2}(C_{2},C_{2})=C_{2} has only one non-trivial class, and
C_{2}⋊C_{2} = C_{2}^{2} | unique split extension with trivial action(direct product) |
C_{2}.C_{2} = C_{4} | unique non-split extension with trivial action. |
n=3. There are two homomorphisms C_{2}→Aut(C_{4})=C_{2}. The trivial one gives
C_{4}⋊C_{2} = C_{2}×C_{4} | unique split extension (direct product) |
C_{4}.C_{2} = C_{8} | unique non-split extension |
C_{4}⋊C_{2} = D_{4} | unique split extension with non-trivial φ |
C_{4}.C_{2} = Q_{8} | unique non-split extension with non-trivial φ |
n>3. There are four homomorphisms
φ: g ↦ 1 | C_{2n-1}⋊_{φ}C_{2} = C_{2n-1}×C_{2} | direct product |
φ: g ↦ -1 | C_{2n-1}⋊_{φ}C_{2} = C_{2n-1}⋊_{1}C_{2} = D_{2n-1} | dihedral group |
φ: g ↦ 2^{n-3}-1 | C_{2n-1}⋊_{φ}C_{2} = C_{2n-1}⋊_{2}C_{2} = SD_{2n} | semidihedral group |
φ: g ↦ 2^{n-3}+1 | C_{2n-1}⋊_{φ}C_{2} = C_{2n-1}⋊_{3}C_{2} = M_{n}(2) | modular maximal-cyclic group |
φ: g ↦1 | C_{2n-1}._{φ}C_{2} = C_{2n} | cyclic group |
φ: g ↦-1 | C_{2n-1}._{φ}C_{2} = Q_{2n} | (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.