つづき

4.Product \(G=AB\): Since \(A \cap B = \{e\}\) (the identity is the only element in \(B\) that fixes 4), the size of the set \(AB\) is:24
Why this happens
For the product \(AB\) to be a group, the factors only need to permute, meaning \(AB = BA\). While having one factor be normal is a sufficient condition for them to permute, it is not a necessary one. In the case of \(S_{4}\), the interaction between the point-stabilizer \(S_{3}\) and the cyclic group \(C_{4}\) allows every element of \(S_{4}\) to be written uniquely as a product \(ab\), even though neither subgroup is "stable" under conjugation by the rest of the group.

https://en.wikipedia.org/wiki/Zappa%E2%80%93Sz%C3%A9p_product
Zappa–Szép product
In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei–Szép product, general product, knit product, exact factorization or bicrossed product) describes a way in which a group can be constructed from two subgroups. It is a generalization of the direct and semidirect products. It is named after Guido Zappa (1940) and Jenő Szép (1950) although it was independently studied by others including B.H. Neumann (1935), G.A. Miller (1935), and J.A. de Séguier (1904).[1]
Internal Zappa–Szép products
Let G be a group with identity element e, and let H and K be subgroups of G. The following statements are equivalent:
G = HK and H ∩ K = {e}
For each g in G, there exists a unique h in H and a unique k in K such that g = hk.
If either (and hence both) of these statements hold, then G is said to be an internal Zappa–Szép product of H and K.
Examples
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External Zappa–Szép products
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(引用終り)
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