Amalgams of \(p\)-groups.

*(English)*Zbl 0246.20015The 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 \(B\).

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 \(B\).

Reviewer: J.Wiegold (MR0167527)

##### 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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[2] | Neumann, H, Generalized free products with amalgamated subgroups II, Am. \(J.\) math., 31, 491-540, (1949) · Zbl 0033.09903 |

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