Unambiguous semidirect products

A semidirect product N ⋊ H is not one group: the expression leaves the action of H on N unsaid, and different actions usually give non-isomorphic groups. Sometimes, though, the factors alone pin the group down — and then N⋊H is a name.

Write N⋊H for the set of groups G with a normal subgroup ≅ N whose complement is ≅ H, and where the split is proper: no complement is normal, so G is not the direct product N × H. The pair (N, H) is an unambiguous splitting when that set holds exactly one group up to isomorphism. That group is what N⋊H names.

• C4⋊C2 is D4, the dihedral group of order 8 — the only proper semidirect product of C4 by C2, since Aut(C4) = C2 offers one non-trivial action.
• C7⋊C9 names the non-abelian group of order 63. The trivial action gives the abelian C63, which is C7 × C9 and keeps its own name.
• C3^2⋊C2 names nothing: inverting both factors of C3 × C3 gives one group of order 18, inverting one factor gives S3 × C3, and two groups cannot share a name.

Why it is unambiguous

Two things make the name determine the group.

• Direct products are excluded. N × H always has its own name, so it is never what N⋊H means. Without that exclusion a name could be ambiguous through no fault of the factors: S3 × C2 has non-normal complements to its S3, so it would answer to S3⋊C2 as well as to its product name.
• Uniqueness is checked against every group of the order. Each G in the set has order |N| · |H|, so the question is decided within that one order — a finite check, made once the order is completely classified. If two groups answer to the same pair, neither is named.

The name is therefore as sharp as a family name like D4 or PSL(2,𝔽7): it picks out one isomorphism class. It is the display name of a group with no family name of its own, and an alias of one that has both — ΓL(1,𝔽8) is also C7⋊C3.

Searching

The search bar accepts N:H as well as N⋊H, with each factor written as any expression that names a group: C7:C9, C3:C4, (C2xC2):C3.

A group may carry several such names, one per normal subgroup that admits an unambiguous splitting, and they all name the same group.

• Schur–Zassenhaus: a normal Hall subgroup N ⊴ G always has a complement, so G = N ⋊ K is a semidirect product with |K| = [G : N], and the complements are all conjugate.