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Amalgams of $$p$$-groups. (English) Zbl 0246.20015
The main aim is the proof of the following elegant theorem. Let $$A\cup B$$ be an amalgam of two finite $$p$$-groups with amalgamation $$U=A\cap B$$. Then $$A\cup B$$ is embeddable in a finite $$p$$-group if there are chief series of $$A$$ and $$B$$ which induce by intersection the same chief series of $$U$$. This has, among other things, the following consequences. (1) If $$U$$ is cyclic, $$A\cup B$$ is embeddable in a finite $$p$$-group. This answers a question put to the author by G. Baumslag. (2) If $$U$$ is normal in $$A$$ and in $$B$$, and $$\operatorname{Aut}_A(U)$$ and $$\operatorname{Aut}_B(U)$$ together generate a $$p$$-group, $$A\cup B$$ is embeddable in a finite $$p$$-group.
The proof of the main theorem is via wreath products, and the principal steps are as follows. Let $$A$$, $$H$$ by any two groups and $$\theta$$ any homomorphism from $$A$$ to $$H$$, with kernel $$X$$. Then a counter-map $$\theta^*$$ is a map from $$H$$ to $$A$$ such that $$(a\theta\cdot h)\theta^*\theta=a\theta\cdot h\theta^*\theta$$ for all $$a$$ in $$A$$, all $$h$$ in $$H$$. Countermaps always exist. To each $$a$$ in $$A$$ define an element $$f_a$$ of the Cartesian power $$X^H$$ by $f_a(h)=[(a\theta\cdot h)\theta^*]^{-1}a\cdot h\theta^*.$ It is readily seen that $$a\to(a\theta,f_a)$$ is an embedding of $$A$$ in the wreath product $$X\wr H$$, which the author terms a standard embedding. There are, of course, very many different standard embeddings. Lemma 1: Let $$U$$ be a subgroup of $$A$$ and $$\theta$$ a homomorphism from $$A$$ to $$H$$ with kernel $$X$$. Given any standard embedding $$u\to(u\theta,f_u)$$ of $$U$$ in $$X\wr H$$, we can find a standard embedding $$a\to(a\theta,g_a)$$ of $$A$$ in $$X\wr H$$ such that $$f_u(h)$$ and $$g_u(h)$$ are conjugate in $$A$$ for all $$u$$ in $$U$$, $$h$$ in $$H$$. This has the important corollary: If $$U\cap X$$ is in the centre of $$A$$, then any standard embedding of $$U$$ in $$X\wr H$$ can be extended to an embedding of $$A$$ in $$X\wr H$$. Lemma 2: Let $$X\cup Y$$ be a normal subamalgam of $$A\cup B$$ (that is, $$X\triangleleft A$$, $$Y\triangleleft B$$, and $$X\cap U=Y\cap U$$), and suppose that $$Z=X\cap U$$ is central in $$A$$ and in $$B$$. If $$X\cup Y$$ can be embedded in $$T$$ and $$(A/X)\cup(B/Y)$$ – with the obvious amalgamation – in $$H$$, then $$A\cup B$$ can be embedded in $$T\wr H$$. The main theorem is then proved by induction on the product $$|A||B|$$ and looking at the subgroups of order $$p$$ in the designated chief series of $$A$$ and $$B$$.
Show Scanned Page ##### MSC:
 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20D15 Finite nilpotent groups, $$p$$-groups 20F14 Derived series, central series, and generalizations for groups
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##### References:
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