Remark. what the models of 2T retain [fgap-001D]
AGENTDRAFTED
The same binary tetrahedral symmetry now has several models. They answer
different questions.
- The concrete model
\(T\leq \mathbb {H}^{\times }\) retains quaternion coordinates, multiplication,
conjugation, and norm. It is the model for explicit calculations.
- The internal semidirect-product structure retains the same
group \(T\), but also marks the subgroups \(Q,C\leq T\). The facts
\[
T=QC,\qquad Q\mathrel {\trianglelefteq }T,\qquad Q\cap C=\{1\}
\]
say that every element of \(T\) has a unique factorization \(qc\). They
expose how the two parts fit inside the concrete group.
- The external model \(Q\rtimes _{\alpha }C\) replaces a
quaternion by a pair \((q,c)\). It retains the factors and records their
interaction in the conjugation action
\(\alpha (c)(q)=cqc^{-1}\). Its multiplication is
\[
(q,c)(q',c')
=\bigl (q\alpha (c)(q'),cc'\bigr ).
\]
- An abstract model can specify the binary tetrahedral group
without quaternion coordinates. This is convenient when only its group
structure is needed, but coordinates and named factors must be restored
by additional maps.
The multiplication map
\[
\mu :Q\rtimes _{\alpha }C\longrightarrow T,\qquad (q,c)\longmapsto qc
\]
is an isomorphism by the internal semidirect-product theorem. Indeed,
\[
\mu \bigl ((q,c)(q',c')\bigr )
=q(cq'c^{-1})cc'
=qcq'c'
=\mu (q,c)\mu (q',c').
\]
Thus the external multiplication is not an analogy: it is exactly the
concrete multiplication written in factor coordinates. The two carriers are
still different, so a theorem or construction passes between them only
through an explicit isomorphism. If it refers to the named factors or their
action, the transport must also record the corresponding compatibility.
The internal and external constructions are developed in
[fre2023discrete, sec. 4.2.13, pp. 62--63] and
[isaev2018theory, sec. 1.4.2, pp. 58--61]. The concrete identification
of \(2T\) with the Hurwitz unit group and
\(Q_8\rtimes \mathbb {Z}/3\mathbb {Z}\) is given in
[voight2021quaternion, sec. 11.2.4, p. 168].