This is the first note of a series dedicated to constructions using amalgamated products and HNN extensions. We begin with an embedding theorem (an alternative proof can be found in our previous note A free group contains a free group of any rank):

Theorem: (Higman-Neumann-Neumann) Any countable group G can be embedded into a two-generator group E. Moreover, it can be supposed that any torsion element of E is conjugated to a element of G.

Proof. Let g_0=1, g_1,g_2, \dots be the elements of G (not necessarily pairwise distinct), and let U= \langle u,v \mid \ \rangle, B= \langle a,b \mid \ \rangle be two copies of the free group of rank two.

We first define the free product A= G \ast U. Clearly,

\{g_0u, g_1vuv^{-1}, g_2 v^2uv^{-2}, \dots \}

freely generates a subgroup H \subset A, because its projection into U is the free subgroup generated by

\{u, vuv^{-1}, v^2uv^{-2}, \dots \}.

In the same way,

\{ a, bab^{-1}, b^2ab^{-2}, \dots \}

freely generates a subgroup K \subset B. Therefore, we may define the amalgamated free product P= A \underset{H=K}{\ast} B by identifying g_iv^iuv^{-i} with b^iab^{-i}. By construction, P is generated by

\{g_0,g_1,g_2, \dots, u,v,a,b \}.

Because g_iv^ivu^{-i}=b^ia b^{-i} and u=g_0u=a, in fact P turns out to be generated by only the three elements \{v,a,b\}.

Then, notice that \langle a,v \rangle and \langle a,b \rangle are two subgroups of P isomorphic to the free group of rank two. Indeed,

\langle a,b \rangle \simeq B and \langle v,a \rangle \simeq \langle v,g_0u \rangle \simeq \langle u,v \rangle \simeq U.

Therefore, if \varphi denotes the isomorphism between \langle a,v \rangle and \langle a,b \rangle sending v to a and a to b, we may define the associated HNN extension E; let t denotes its stable letter. Now E is generated by \{v,a,b,t \}, but tvt^{-1}=a and tat^{-1}=b, so E is in fact generated by \{v,t\}.

Consequently, we just embedded G into the two-generator group E. Moreover, we notice that a torsion element of E is conjugated to an element of G because a torsion element of an amalgamated product or of a HNN extension is necessarily conjugated to an element of a factor (and because U and B are torsion-free). \square

As a corollary, we deduce the following theorem that we already proved in the note Cantor-Bendixson rank in group theory: A theorem of B.H. Neumann, using the space of marked groups and small cancellation groups.

Theorem: (B.H. Neumann) There exist exactly 2^{\aleph_0} nonisomorphic two-generator groups.

Proof. First, because any two-generator group is a quotient of a free group of rank two, there are at most 2^{\aleph_0} nonisomorphic two-generator groups. To conclude, it is sufficient to exhibit a family of 2^{\aleph_0} nonisomorphic two-generator groups.

Let p_1 < p_2 < \cdots denote the sequence of consecutive primes. For any subset I \subset \mathbb{N}, we define

A_I = \bigoplus\limits_{i \in I} \mathbb{Z}_{p_i},

and G_I the two-generator group associated to A_I given by the previous theorem.

If I \neq J, say there exists i \in I \backslash J, A_I (and a fortiori G_I) has an element of order p_i; however, A_J (and a fortiori G_J) don’t. Therefore, G_I and G_J are not isomorphic. \square