Remark. internal, external, and abstract models [fgap-000L]
Remark. internal, external, and abstract models [fgap-000L]
Three related objects have appeared. The internal model \(T\leq \mathbb {H}^{\times }\) has quaternions as its elements. The external semidirect product \(Q\rtimes _{\alpha }C\) has pairs \((q,c)\) as its elements. An abstract binary tetrahedral group can instead be specified without quaternion coordinates.
The isomorphism \((q,c)\mapsto qc\) compares the first two models. It is not an equality of their elements.
The next note transports the distinguished central involution and
cardinality to an abstract binary tetrahedral model. Transporting the full
factorization to factors such as QuaternionGroup 2 and
Multiplicative (ZMod 3) also needs explicit isomorphisms and
equivariance proofs. None of this follows from the order computation
\(|T|=24\) alone.