Finite quotients of residually approximating groups and finite traces of group varieties

Problem

Characterize the classes $\mathcal K$ of finite groups that occur as all finite members of an ordinary group variety:

\mathcal K=\mathcal V_{\mathrm{fin}} =\{Q\in\mathcal V:|Q|<\infty\}.

The criterion below uses finite quotients of finitely generated residually- $\mathcal K$ groups. Residual separation allows arbitrary homomorphisms into members of $\mathcal K$ .

The residual criterion

An ordinary group variety is a class defined by word identities in the group signature. For such a variety $\mathcal V$ , write

\mathcal V_{\mathrm{fin}}=\{Q\in\mathcal V:|Q|<\infty\}.

Problem 4.17 of the Kourovka Notebook, due to R. Baer, asks for a characterization of the classes $\mathcal V_{\mathrm{fin}}$ (2).

Let $\mathcal K$ be a class of finite groups. A group $G$ is residually- $\mathcal K$ if

\tag{2} \forall g\in G\setminus\{1\}\quad \exists Q\in\mathcal K\quad\exists f:G\longrightarrow Q \quad f(g)\ne1.

All maps between groups in this article are homomorphisms. The maps in (2) need not be surjective. Consider the condition

\tag{R} \left. \begin{gathered} G\text{ is finitely generated and residually-}\mathcal K,\\ p:G\twoheadrightarrow Q,\qquad |Q|<\infty \end{gathered} \right\}\quad\Longrightarrow\quad Q\in\mathcal K.

Theorem 1. For a class $\mathcal K$ of finite groups, the following are equivalent:

$\mathcal K=\mathcal V_{\mathrm{fin}}$ for an ordinary group variety $\mathcal V$ ;
$\mathcal K$ satisfies (R).

When these conditions hold, one may take $\mathcal V$ to be the class of all groups satisfying every word identity valid in $\mathcal K$ .

Here finite generation means the existence of an epimorphism $F_n\twoheadrightarrow G$ , where $F_n=F(x_1,\ldots,x_n)$ and $n\ge0$ . In particular, the trivial group is allowed. No closure hypothesis on $\mathcal K$ is imposed. In the implication (1) $\Rightarrow$ (2), finite generation of $G$ is unnecessary.

Common laws and residual separation

Regard an identity as a pair $(n,w)$ with $w\in F_n$ ; its assertion in $G$ is

\tag{4} G\models(n,w)\quad\Longleftrightarrow\quad f(w)=1\quad\text{for every }f:F_n\longrightarrow G.

This includes all ordinary group equations: the equation $u=v$ is written as the identity $uv^{-1}=1$ . For a set $\Sigma$ of such pairs, let $\operatorname{Mod}(\Sigma)$ denote the groups satisfying every member of $\Sigma$ .

Lemma 2. Let $\Sigma$ be a set of ordinary identities.

If every member of $\mathcal K$ satisfies $\Sigma$ , then every residually- $\mathcal K$ group satisfies $\Sigma$ .
If $G$ satisfies $\Sigma$ and $p:G\twoheadrightarrow H$ , then $H$ satisfies $\Sigma$ .

Proof. For the first assertion, fix $(n,w)\in\Sigma$ and $f:F_n\to G$ . If $f(w)\ne1$ , residual separation gives $Q\in\mathcal K$ and $q:G\to Q$ with $qf(w)\ne1$ . This contradicts $Q\models(n,w)$ .

For the second assertion, let $f:F_n\to H$ . Choose $g_i\in G$ with $p(g_i)=f(x_i)$ , and let $\widetilde f:F_n\to G$ send $x_i$ to $g_i$ . The universal property of $F_n$ gives $p\widetilde f=f$ . Consequently,

f(w)=p\bigl(\widetilde f(w)\bigr)=p(1)=1 \qquad ((n,w)\in\Sigma).

◻

For $n\ge0$ , define

\tag{6} R_n(\mathcal K)= \bigcap_{\substack{Q\in\mathcal K\\ f:F_n\to Q}}\ker f, \qquad L_n(\mathcal K)=F_n/R_n(\mathcal K).

The intersection is a normal subgroup of $F_n$ ; an empty intersection means $F_n$ . Put

\tag{7} \Sigma_{\mathcal K} =\{(n,w):n\ge0,\ w\in R_n(\mathcal K)\}.

Thus $\Sigma_{\mathcal K}$ is exactly the set of identities common to $\mathcal K$ . In particular,

\tag{8} Q\in\mathcal K\quad\Longrightarrow\quad Q\models\Sigma_{\mathcal K}.

Lemma 3. The group $L_n(\mathcal K)$ is finitely generated and residually- $\mathcal K$ .

Proof. The images of $x_1,\ldots,x_n$ generate $L_n(\mathcal K)$ . Let $wR_n(\mathcal K)\ne1$ . By (6), there are $Q\in\mathcal K$ and $f:F_n\to Q$ such that $f(w)\ne1$ . Since $R_n(\mathcal K)\le\ker f$ , the map $f$ induces

\overline f:L_n(\mathcal K)\longrightarrow Q, \qquad \overline f\bigl(wR_n(\mathcal K)\bigr)=f(w)\ne1.

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Lemma 4. If $G\models\Sigma_{\mathcal K}$ and $p:F_n\twoheadrightarrow G$ , then $p$ factors through an epimorphism $L_n(\mathcal K)\twoheadrightarrow G$ .

Proof. For $w\in R_n(\mathcal K)$ , the pair $(n,w)$ belongs to $\Sigma_{\mathcal K}$ , so (4) gives $p(w)=1$ . Hence $R_n(\mathcal K)\le\ker p$ , and the quotient universal property yields

\overline p:L_n(\mathcal K)\longrightarrow G, \qquad \overline p\bigl(wR_n(\mathcal K)\bigr)=p(w).

Every $g\in G$ is $p(w)$ for some $w\in F_n$ , proving surjectivity. ◻

Proof of the characterization

Proof of Theorem 1. Suppose $\mathcal K=\operatorname{Mod}(\Sigma)_{\mathrm{fin}}$ . Let $G$ be residually- $\mathcal K$ and let $p:G\twoheadrightarrow Q$ , where $Q$ is finite. By Lemma 2(1), $G\models\Sigma$ ; by Lemma 2(2), $Q\models\Sigma$ . Therefore $Q\in\mathcal K$ , proving (R).

Conversely, suppose (R) holds and set $\mathcal V=\operatorname{Mod}(\Sigma_{\mathcal K})$ . Inclusion $\mathcal K\subseteq\mathcal V_{\mathrm{fin}}$ is (8). If $Q\in\mathcal V_{\mathrm{fin}}$ , choose a finite generating tuple and its associated epimorphism $F_n\twoheadrightarrow Q$ . Lemma 4 produces

L_n(\mathcal K)\twoheadrightarrow Q.

Lemma 3 shows that its domain is finitely generated and residually- $\mathcal K$ . Condition (R) now gives $Q\in\mathcal K$ . Thus

\mathcal K=\operatorname{Mod}(\Sigma_{\mathcal K})_{\mathrm{fin}},

as asserted. ◻

The quotient construction (6) is the group instance of the free-algebra construction in Birkhoff’s variety theorem (1).

References

Preprint · Lean (GitHub)

G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc. 31 (1935), 433–454. doi:10.1017/S0305004100013463.

E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st ed., 2026, Problem 4.17. arXiv:1401.0300.