Small partial Latin squares that embed in an infinite group but not into any finite group

TL;DR

This paper determines the minimal size of small partial Latin squares that can embed in an infinite group but not in any finite group, providing explicit examples and leveraging computational methods.

Contribution

It proves the minimal size is 12, identifies 50 such examples up to equivalence, and shows they embed in the Baumslag group, advancing understanding of embeddings in group theory.

Findings

Minimal size of such partial Latin squares is 12

50 examples of size 12 exist up to equivalence

All examples embed in the Baumslag group

Abstract

Suppose that $Y_1,Y_2,Y_3$ are finite sets and $P\subseteq Y_1\times Y_2\times Y_3$. We say that $P$ embeds in a group $G$ if there exist injective maps $\phi_i\colon Y_i\rightarrow G$ for $i=1,2,3$ such that $\phi_1(y_1)\phi_2(y_2)=\phi_3(y_3)$ for each $(y_1,y_2,y_3)\in P$. Hirsch and Jackson asked for the cardinality of the smallest $P$ that embeds in some infinite group but not into any finite group. We prove that the answer to their question is 12. Moreover, we show that there are 50 examples of cardinality 12, up to equivalence, and each of them embeds in the (infinite) Baumslag group $G=\langle a,b \mid b=[b,b^a]\rangle$. Our proof uses computations to answer questions about finitely presented groups which are known to be algorithmically undecidable in general.

Tables

Table 1. Table 1: Number of species of candidate PLS of size at most 12.

size	1	2	3	4	5	6	7	8	9	10	11	12
all	1	2	5	18	59	306	1861	15097	146893	1693416	22239872	327670703
conn.	1	1	3	11	36	213	1405	12274	125235	1490851	20003121	299274006
cand.	0	0	0	2	0	11	50	489	6057	92533	1517293	27056665

Table 2. Table 2: Counts of species of candidate PLS which embed in certain groups

size	NE	abelian	nonabelian	infNotFin
4	0	2	0	0
6	0	10	1	0
7	2	44	4	0
8	16	435	38	0
9	147	5447	463	0
10	2402	82555	7576	0
11	42884	1338816	135593	0
12	854559	23520406	2681650	50

Equations

$${\varphi }_1:\{1,2,\dots ,m\}\to G,{\varphi }_2:\{1,2,\dots ,n\}\to G,{\varphi }_3:\Lambda (P)\to G,$$

$$\mathcal{R}=\{R_i{C}_j{P}_{i,j}^{-1}\in F:(i,j)\in \mathcal{S}(P)\}.$$

$$\mathcal{M}=\{r_i{r}_j^{-1}:1⩽i<j⩽m\}\cup \{c_i{c}_j^{-1}:1⩽i<j⩽n\}\cup \{s_i{s}_j^{-1}:1⩽i<j⩽k\}.$$

$$P=\begin{array}[]{|c|c|c|c|c|}\hline\cr&b&&c&\\ \hline\cr a&&c&&\\ \hline\cr&a&b&&\\ \hline\cr b&&&&d\\ \hline\cr c&d&&&\\ \hline\cr&&&b&c\\ \hline\cr\end{array}$$

$$r_2$$

$$r_3$$

$$r_4$$

$$r_5$$

$$r_6$$

$$\begin{array}[]{|c||c|c|c|c|c|}\hline\cr&1&c_{2}&c_{3}&c_{4}&c_{5}\\ \hline\cr\hline\cr 1&&b&&c&\\ \hline\cr r_{2}&a&&c&&\\ \hline\cr r_{3}&&a&b&&\\ \hline\cr r_{4}&b&&&&d\\ \hline\cr r_{5}&c&d&&&\\ \hline\cr r_{6}&&&&b&c\\ \hline\cr\end{array}$$

$$\phi :⟨\hat{X}\mid \hat{\mathcal{R}}⟩\to ⟨P⟩,$$

$$X=\{R_i,C_j:(i,j)\in \mathcal{S}(P)\}\cup \Lambda (P)$$

$$P=\begin{array}[]{|c|c|c|c|}\hline\cr a&d&&\\ \hline\cr&a&d&\\ \hline\cr&b&&c\\ \hline\cr c&&b&a\\ \hline\cr\end{array}$$

$$\mathcal{R}=\{a,C_2{d}^{-1},R_2{C}_2{a}^{-1},R_2{C}_3{d}^{-1},R_3{C}_2{b}^{-1},R_3{C}_4{c}^{-1},R_4{c}^{-1},R_4{C}_3{b}^{-1},R_4{C}_4{a}^{-1}\}.$$

$$P=\begin{array}[]{|c|c|c|c|}\hline\cr a&b&c&\\ \hline\cr b&d&&c\\ \hline\cr c&&d&\\ \hline\cr&e&&a\\ \hline\cr\end{array},$$

$$x$$

$$y$$

$$z$$

$$\begin{array}[]{|c|c|c|c|c|}\hline\cr a&b&&&\\ \hline\cr b&a&&&\\ \hline\cr&&c&d&e\\ \hline\cr&&d&e&c\\ \hline\cr&&e&c&d\\ \hline\cr\end{array}$$

$$\{1⩽i⩽\ell :c_i\in {\Pi }_2(T^{\prime })\}\cap \{1⩽i⩽\ell :s_i\in {\Pi }_3(T^{\prime })\}=\varnothing .$$

$$B_1=⟨a,b\mid b=[b,(b^{-2}{)}^a]⟩\text{or}{B}_2=⟨a,b\mid b=[b,(b^2{)}^a]⟩,$$

$$P_{1}=\begin{array}[]{|c|c|c|c|c|c|}\hline\cr a&b&c&d&&\\ \hline\cr b&e&&&c&\\ \hline\cr&&e&&a&d\\ \hline\cr&&&e&&a\\ \hline\cr\end{array}\hskip 56.9055ptP_{2}=\begin{array}[]{|c|c|c|c|c|}\hline\cr a&b&c&&\\ \hline\cr b&d&&c&\\ \hline\cr e&&&&d\\ \hline\cr&&d&a&\\ \hline\cr&&&e&a\\ \hline\cr\end{array}$$

Full text

1 Introduction and results

We define a partial Latin square (PLS) $P$ to be a matrix in which some cells may be empty and in which each filled cell contains one symbol from an underlying alphabet $\Lambda(P)$, such that no symbol occurs more than once within any row or column. The size of $P$ is the number of filled cells in $P$, and the shape $\mathcal{S}(P)$ of $P$ is the set of filled cells in $P$. We avoid degeneracies by insisting that each row and column contains at least one filled cell, and that each element of $\Lambda(P)$ appears at least once in $P$. A side-effect is that our PLS need not be square matrices. In some references, PLS are defined to be square matrices and to have at least as many rows as there are symbols. To achieve these properties it is always possible to add empty rows and/or empty columns to our PLS. Allowing any finite number of empty rows and columns would create some technical nuisances but would not materially affect any of the questions we are interested in solving.

Let $P=[P_{i,j}]$ be an $m\times n$ PLS. An embedding $\phi\colon P\hookrightarrow G$ of $P$ into a group $G$ is a triple $(\phi_{1},\phi_{2},\phi_{3})$ of injective maps

[TABLE]

such that $\phi_{1}(i)\phi_{2}(j)=\phi_{3}(P_{i,j})$ for all $(i,j)\in\mathcal{S}(P)$. We refer to $r_{i}=\phi_{1}(i)$ for $i=1,\dots,m$ as the row labels, and $c_{j}=\phi_{2}(j)$ for $j=1,\dots,n$ as the column labels. Intuitively, $P$ embeds in $G$ if a copy of $P$ can be found within the Cayley table of $G$ (in the subtable which has row and column labels $r_{1},\ldots,r_{m}$ and $c_{1},\ldots,c_{n}$, respectively). We refer to [3] for a discussion of applications of embeddings of PLS in groups and connections with linear algebra and topological graph theory. Note that embedding in groups is a special type of completion problem for PLS. See [5, Chap.3] for an introduction to the rich literature of such problems.

A useful notion of equivalence for PLS is obtained by converting each PLS $P$ into a set of triples $\mathcal{T}(P)=\{(r,c,P_{r,c}):(r,c)\in\mathcal{S}(P)\}$. We say that two PLS are from the same species if they produce the same set of triples, modulo uniform permutation of the 3 coordinates in the triples, and relabelling within each coordinate. The property of having an embedding in a given group is a species invariant [3, Lem.1], so in this paper it will suffice to consider one representative from each species of PLS.

Hirsch and Jackson [7, Ex.3.7] gave an example of a PLS of size 29 that can be embedded in an infinite group, but not in any finite group. They noted that smaller examples exist, and posed the question of what the smallest possible size of such a PLS is. The purpose of this paper is to answer this question by proving:

Theorem 1**.**

There are $50$ species of PLS of size $12$ that can be embedded in an infinite group, but not in any finite group. No PLS of smaller size has the same property.

One of the 50 PLS of size $12$ is detailed in Lemma 7. The remainder of the paper discusses our computational approach which proves Theorem 1. These computations involve manipulating finite presentations for groups to test, for example, whether a group is finite or whether two group elements are equal. These questions are equivalent to solving the “word problem” and hence, in full generality, are well known to be algorithmically undecidable, cf. [8, §5] or [12, p.54]. So we were somewhat fortunate to find methods which solved the instances of these problems that we needed to solve.

For recent work related to this paper we refer to [13], which identifies all smallest PLS that (a) do not embed into any group, (b) embed into a group but do not embed into any abelian group, or (c) embed into an abelian group but do not embed into any cyclic group. Each of those classes contains small PLS that can be found with simpler methods than we employ in the present paper.

2 The presentation defined by a PLS

The next lemma (cf. [3, Lem.2]) is an easy observation and shows that we lose no generality by assuming that the associated row and column labels of an embedding satisfy $r_{1}=c_{1}=1$, the identity of the group.

Lemma 2**.**

If a PLS $P$ can be embedded into a group $G$, then it can be embedded with row labels $r_{1},\ldots,r_{m}\in G$ and column labels $c_{1},\ldots,c_{n}\in G$ that satisfy $r_{1}=c_{1}=1$, the identity in $G$.

Proof.

If $(\phi_{1},\phi_{2},\phi_{3})\colon P\hookrightarrow G$ is an embedding with associated row labels $r_{1},\ldots,r_{m}$ and column labels $c_{1},\ldots,c_{n}$, then $(\phi^{\prime}_{1},\phi^{\prime}_{2},\phi^{\prime}_{3})\colon P\hookrightarrow G$ defined by $\phi_{1}^{\prime}(r)=r_{1}^{-1}\phi_{1}(r)$, $\phi_{2}^{\prime}(c)=\phi_{2}(c)c_{1}^{-1}$, and $\phi_{3}^{\prime}(e)=r_{1}^{-1}\phi_{3}(e)c_{1}^{-1}$ is an embedding with the required property. ∎

Let $P$ be an $m\times n$ PLS. Let $F$ be the free group on $X=\{R_{i},C_{j}:(i,j)\in\mathcal{S}(P)\}\cup\Lambda(P)$ and let

[TABLE]

The presentation defined by $P$ is $\{X\mid\mathcal{R}\}$ and the group defined by $P$ is $\langle P\rangle=\langle X\mid\mathcal{R}\rangle$.

Lemma 3**.**

Let $P$ be an $m\times n$ PLS, and let $(\phi_{1},\phi_{2},\phi_{3})\colon P\hookrightarrow G$ be an embedding into a group $G$, with associated row and column labels $r_{1},\ldots,r_{m}$ and $c_{1},\ldots,c_{n}$, respectively. Then $P$ embeds in the subgroup $H$ of $G$ generated by $\{r_{i},c_{j},\phi_{3}(P_{i,j}):(i,j)\in\mathcal{S}(P)\}$, and $H$ is a quotient of $\langle P\rangle$. In particular, $P$ embeds into $\langle P\rangle$.

Proof.

Clearly, $P$ embeds into the group $H$, and the generators of $H$ satisfy $r_{i}c_{j}=\phi_{3}(P_{i,j})$ for all $(i,j)\in\mathcal{S}(P)$. By von Dyck’s Theorem [12, 2.2.1], the map $X\to H$ defined by $R_{i}\mapsto r_{i}$, $C_{j}\mapsto c_{j}$, $P_{i,j}\mapsto\phi_{3}(P_{i,j})$ for all $(i,j)\in\mathcal{S}(P)$ extends to a surjective group homomorphism $\langle P\rangle\to H$. ∎

In later sections we describe how we used the computer algebra system GAP [6] to answer questions about possible embeddings of PLS. Typically, Lemma 3 provides the starting point for these investigations, since it says that if $P$ is going to embed in any group, it must embed in $\langle P\rangle$. In the remainder of this section we explore some of consequences of finding an embedding.

Recall that a group $G$ is residually finite if for every nontrivial $g\in G$ there is a normal subgroup $N\unlhd G$ with $g\notin N$ such that $G/N$ is finite.

Lemma 4**.**

Let $P$ be a PLS such that $\langle P\rangle$ is a residually finite group. If $P$ can be embedded into $\langle P\rangle$, then $P$ can be embedded into a finite group.

Proof.

Let $\phi=(\phi_{1},\phi_{2},\phi_{3})\colon P\hookrightarrow\langle P\rangle$ be an embedding, with associated row and column labels $r_{1},\ldots,r_{m}$ and $c_{1},\ldots,c_{n}$, respectively. Let $\{s_{1},\ldots,s_{k}\}=\{\phi_{3}(s):s\in\Lambda(P)\}$ be the set of symbols in the embedded PLS. Define

[TABLE]

Note that each $r\in\mathcal{M}$ is nontrivial since $\phi$ is an embedding. By assumption, for each $r\in\mathcal{M}$ there exists a normal subgroup $U_{r}<\langle P\rangle$ of finite index with $r\notin U_{r}$. Since the intersection of two finite-index subgroups has finite index, see [12, 1.3.11], it follows that $U=\bigcap_{r\in\mathcal{M}}U_{r}$ is a normal subgroup of $\langle P\rangle$ of finite index, that is, $H=\langle P\rangle/U$ is finite. By construction, $r\notin U$ for all $r\in\mathcal{M}$. Thus, if $\pi\colon\langle P\rangle\to H$ is the natural epimorphism, then $\phi^{\prime}=(\pi\circ\phi_{1},\pi\circ\phi_{2},\pi\circ\phi_{3})$ embeds $P$ into $H$. ∎

Note that every free group is residually finite, see [12, 6.1.9]; this yields the following corollary.

Corollary 5**.**

Let $P$ be a PLS such that $\langle P\rangle$ is a free group. If $P$ can be embedded into $\langle P\rangle$, then $P$ can be embedded into a finite group.

Lemma 6**.**

Let $P$ be a PLS. If $P$ can be embedded into the abelianisation $\langle P\rangle/\langle P\rangle^{\prime}$, then $P$ can be embedded into a finite abelian group.

Proof.

Write $G=\langle P\rangle$ and $A=G/G^{\prime}$. Let $(\phi_{1},\phi_{2},\phi_{3})\colon P\hookrightarrow A$ be an embedding, with row and column labels $r_{1},\ldots,r_{u}$ and $c_{1},\ldots,c_{v}$, respectively. The fundamental theorem of finitely generated abelian groups [12, 4.2.10] shows that $A\cong Z_{1}\times\cdots\times Z_{k}\times F$ for some finite group $F$ and integer $k\geqslant 0$, where each $Z_{i}\cong\mathbb{Z}$ is a infinite cyclic group with generator $g_{i}$. Thus each label $r_{i}$, $c_{j}$, and each symbol in $\phi_{3}(\Lambda(P))$ can be written as $g_{1}^{e_{1}}\cdots g_{k}^{e_{k}}f$ for certain $e_{1},\ldots,e_{k}\in\mathbb{Z}$ and $f\in F$. Let $\mathcal{E}_{i}$ be the set of exponents of $g_{i}$ occurring in all these expressions. Now choose an integer $m_{i}$ such that $m_{i}>2|e|$ for all $e\in\mathcal{E}_{i}$. Then $B=A/\langle g_{1}^{m_{1}},\ldots,g_{k}^{m_{k}}\rangle$ is a finite abelian group, and, by construction, $P$ can be embedded into $B$. ∎

3 An example of size 12

In this section we explicitly present a PLS of size 12 that cannot be embedded in any finite group but can be embedded in an infinite group. This will turn out to be the smallest possible size for such an example.

Lemma 7**.**

The PLS

[TABLE]

can be embedded in an infinite group, but in no finite group.

Proof.

In [1], Baumslag defined the finitely presented group $G=\langle A,B\mid B=[B,B^{A}]\rangle$, where, as usual, $B^{A}=A^{-1}BA$ and $[B,B^{A}]=B^{-1}B^{(B^{A})}$. He showed that $G$ is infinite, non-cyclic, and $B=1$ in every finite quotient of $G$. We use these properties to prove the claim.

First, suppose we have an embedding $(\phi_{1},\phi_{2},\phi_{3})\colon P\hookrightarrow H$ into some group $H$, with row and column labels $r_{1},\ldots,r_{6}$ and $c_{1},\ldots,c_{5}$, respectively. For simplicity, we identify $a,b,c,d$ with their images under $\phi_{3}$, and, by Lemma 2, we assume that $r_{1}=c_{1}=1$. We deduce from Figure 1 that

[TABLE]

Hence, $bab^{2}a^{-1}b=r_{4}c_{5}=d=aba^{-1}b^{2}$, that is, $bab^{2}a^{-1}=aba^{-1}b$, and so $b^{2}=a^{-1}b^{-1}aba^{-1}ba$ and $b=b^{-1}a^{-1}b^{-1}aba^{-1}ba$, which can be written as $b=[b,b^{a}]$. This shows that the subgroup $K=\langle a,b\rangle$ of $H$ satisfies $b=[b,b^{a}]$. By von Dyck’s Theorem, $K$ is an epimorphic image of Baumslag’s group $G=\langle A,B\rangle$ with $A,B\in G$ mapped to $a,b\in K$, respectively. If $H$ is finite, then so is $K$, which forces $b=1$; so $r_{4}=b=1=r_{1}$, contradicting the injectivity of $\phi_{1}$. We conclude that $P$ cannot be embedded in a finite group.

It remains to prove that $P$ embeds into the infinite Baumslag group $G$. Since the group defined by $P$ is isomorphic to $G$, we simply choose the row and column labels (and symbols) as in (1), replacing $a$ and $b$ by $A$ and $B$, respectively. Since $G$ is non-cyclic by [1], it is an easy exercise to verify that the row labels are pairwise distinct, and similarly for the column labels and the symbols. For example, if $r_{3}=r_{5}$, then $B^{-1}=BA^{-1}B$, and so $B^{-3}=A^{-1}$, showing $\langle A,B\rangle$ is cyclic. The only nontrivial cases to consider are $r_{4}=r_{6}$, $r_{5}=r_{6}$, $c_{2}=c_{5}$ and $b=d$, since for these it is necessary to use the defining relation $B=[B,B^{A}]$ as well. For example, it follows from $r_{5}=r_{6}$ that $BA^{-1}B=B^{-1}A^{-1}$; substituting this in $B=[B,B^{A}]=B^{-1}A^{-1}B^{-1}ABA^{-1}BA$, one obtains $B=B^{-1}A^{-1}B^{-1}AB^{-1}$. This yields $B^{A}=B^{-3}$, and now it follows from $1=[B,B^{A}]=B$ that $\langle A,B\rangle$ is cyclic, again yielding a contradiction. ∎

4 Computational methods

To guarantee that we have found a PLS of smallest possible size which embeds into an infinite group, but in no finite group, we have to show that every PLS of smaller size either embeds in some finite group, or does not embed in any group. In this section we describe some computational methods which help to test these properties.

Throughout this section, let $P$ be an $m\times n$ PLS, with associated presentation $\{X\mid\mathcal{R}\}$ and group $\langle P\rangle=\langle X\mid\mathcal{R}\rangle$. If $P$ can be embedded in some group $H$, then $H$ is a quotient of $\langle P\rangle$, and $P$ also embeds in $\langle P\rangle$, see Lemma 3. Therefore our aim is to find an embedding of $P$ in some finite quotient of $\langle P\rangle$, or to show that $P$ cannot be embedded in any group. The latter happens, for example, if $\langle P\rangle$ is the trivial group. As mentioned in the introduction, it is in general an algorithmically undecidable problem to determine whether a finitely presented group is trivial or finite. Below we comment on our approaches, which avoided this difficulty.

4.1 Proving that a PLS cannot be embedded into any group

Using the relations $\mathcal{R}$ of $\langle P\rangle$, we can reduce the number of generators of $\langle P\rangle$. More precisely, we use Tietze transformation (see [8, §2.4.4]) to obtain a second presentation $\{\hat{X}\mid\hat{\mathcal{R}}\}$ of $\langle P\rangle$, which usually has significantly fewer generators and relations. We call this presentation a reduced presentation defined by $P$. In the examples we have considered, that is, PLS of size at most 12, there are usually one to three generators in $\hat{X}$ and one to three relations in $\hat{\mathcal{R}}$. We can use GAP to do these Tietze transformations, to construct an explicit isomorphism

[TABLE]

and to write the original generators

[TABLE]

as words in the new generators $\hat{X}$. After this rewriting, it often becomes visible that there are duplicates in the list $(R_{1},\ldots,R_{m})$, in $(C_{1},\ldots,C_{n})$, or in $\Lambda(P)$, each considered as a list of elements in the free group generated by $\hat{X}$. If this is the case, then we have established that $P$ cannot be embedded into $\langle P\rangle$. Since any embedding of $P$ into some group $H$ implies that $H$ is a quotient of $\langle P\rangle$, this proves that $P$ cannot be embedded in any group. Example 8 illustrates this with an explicit PLS.

Example 8**.**

The PLS

[TABLE]

yields a presentation $\{X\mid\mathcal{R}\}$ where $X=\{R_{2},R_{3},R_{4},C_{2},C_{3},C_{4},a,b,c,d\}$ and

[TABLE]

Note that we can assume $R_{1}=C_{1}=1$ by Lemma 2. We construct $G=\langle P\rangle$ with the GAP code in Figure 2. Applying Tietze transformations reveals that $G\cong\mathbb{Z}$ is a one-generator group with no relations. We then express $X$ in terms of the new generator and obtain that $c=d$ in $G$. This shows that $P$ cannot be embedded in any group: in any such embedding, we would have $c=d$, a contradiction.

What we have described is a very fast test to identify some PLS which cannot be embedded into any group: this is possible because our check for duplicates (in the lists of symbols, row labels, and column labels) is in the free group generated by $\hat{X}$, where the word problem is trivial.

If this test does not establish that we cannot embed the given PLS in any group (and our attempts to find an embedding fail, cf. the approaches described below), then we proceed as follows. Again, let $\hat{X}$ and $\hat{\mathcal{R}}$ be as above. We use the Knuth-Bendix completion algorithm (provided by the GAP package KBMAG, see also [8, §12.2]) to construct from $\hat{\mathcal{R}}$ a rewriting system which can be used to rewrite an element of the group $\langle P\rangle=\langle\hat{X}\mid\hat{\mathcal{R}}\rangle$ in so-called reduced form. If this rewriting system happens to be confluent, then two elements in $\langle P\rangle$ are identical if and only if their reduced forms coincide. If the rewriting system is not confluent, then it can still happen that two different reduced forms correspond to the same element. In all our examples, this method was sufficient to prove that a PLS which we failed to embed into some finite group can in fact not be embedded into any group. We illustrate this in Example 9.

Example 9**.**

Starting with the PLS

[TABLE]

GAP finds a reduced presentation $\{u,v\mid v^{2}u^{-2}\}$ for $\langle P\rangle$ with row labels, column labels, and symbols as given in Figure 3. Considering the elements in these lists in the free group on $\{u,v\}$, GAP cannot detect duplicates. However, after applying the Knuth-Bendix algorithm, we identify duplicates among the lists of reduced forms; this proves that $P$ cannot be embedded in any group.

4.2 Proving that a PLS can be embedded into an abelian group

Suppose that $P$ can be embedded into an abelian group $H$. Since $H$ is a quotient of $\langle P\rangle$, this embedding lifts to an embedding $\phi\colon P\hookrightarrow A$ into the abelianisation $A=\langle P\rangle/\langle P\rangle^{\prime}$ of $\langle P\rangle=\langle X\mid\mathcal{R}\rangle$; note that $A=\langle X\mid\mathcal{R}\cup\{[a,b]:a,b\in X\}\rangle$. On the other hand, we have seen in Lemma 6 that any such embedding gives rise to an embedding of $P$ into a finite abelian group. It is straightforward to check whether $P$ can be embedded in $A$: if this is the case, then $P$ can be embedded into a finite abelian group; if not, then we have proved that $P$ cannot be embedded in any abelian group.

4.3 Finding an embedding into a nonabelian finite group

Let $P$ be as before, and suppose the method described in §4.2 shows that $P$ cannot be embedded into an abelian group. Also, suppose that the method in §4.1 did not conclude that $P$ fails to embed in any group. We now want to check whether $P$ can be embedded into a finite nonabelian group. Recall that if $\phi\colon P\hookrightarrow H$ is such an embedding, then $H$ is a finite quotient of $\langle P\rangle$. As before, we first apply Tietze transformations to find a shorter presentation of $\langle P\rangle$, say $\langle P\rangle\cong\langle\hat{X}\mid\hat{\mathcal{R}}\rangle$. We then use several quotient algorithms and brute-force techniques to look for an embedding.

Brute-force techniques I. Using the small generating set $\hat{X}$, we can simply enumerate all possible homomorphisms from $\langle P\rangle$ into some small finite group $H$, say of order at most 24. We check if any of these homomorphisms yields an embedding of $P$.

Nilpotent quotients. We use the nilpotent quotient algorithm in GAP (provided by the GAP package NQ, see also [8, §9.4.3]) to compute the largest quotients of $\langle P\rangle$ having nilpotency class $c=1,2,3,\ldots$. (For $c=1$ this quotient is the abelianisation of $\langle P\rangle$.) If $P$ embeds in one of these quotients, say $K_{c}=\langle P\rangle/\gamma_{c+1}(\langle P\rangle)$, then we try to embed $P$ into a finite quotient of $K_{c}$, by adding several random relations. If $P$ does not embed into $K_{c}$, then $P$ does not embed in any nilpotent group of class $c$. Note that one can very efficiently compute with finitely presented nilpotent groups.

Low Index Subgroups. Based on the work in [9], there exist efficient methods to compute with finite-index subgroups of finitely presented groups. For example, all subgroups of $\langle P\rangle$ of index at most $i$, where $i$ is some small integer, say $i\leqslant 6$, can be computed, up to conjugacy, with the GAP command LowIndexSubgroupsFpGroup. As explained in [9], these subgroups are stored efficiently so that one can compute their intersection $U$, and a permutation representation of $\langle P\rangle$ on the cosets of $U$. This yields a homomorphism from $\langle P\rangle$ into the symmetric group of degree $[\langle P\rangle:U]$, the index of $U$ in $\langle P\rangle$. We can readily check whether $\langle P\rangle$ embeds in this finite symmetric group.

We note that, in our examples, the previous three techniques (and the methods described in §4.1 and §4.2) were sufficient to deal with all occurring PLS. In practice, the Low Index Subgroup method was only required to deal with certain PLS of size 12. Below we list one other approach which could be used to consider other examples.

Brute-force techniques II. If the number of generators in $\hat{X}$ is larger than the number of relations in $\hat{\mathcal{R}}$, then $\langle P\rangle$ is infinite. In this case, we add some random relations to $\hat{\mathcal{R}}$ until the number of relations is at least the number of generators. We then try to compute the order of the new group using coset-enumeration techniques (see [8, §5]). If the number of cosets used becomes too big, we abort this computation. (Unfortunately, the number of cosets used is not immediately related to the actual order of the group.) If we determine that the group is finite, then we check whether we have found an embedding of $P$.

5 The candidates

Our mission is to find all of the smallest PLS that embed in an infinite group, but do not embed in any finite group. On the basis of the example given in §3, we only need to consider PLS of size at most $12$. Also, as mentioned in the introduction, it suffices to consider one representative from each species of PLS. Nevertheless, there are a great many species. So, in this section, we discuss techniques which allow us to prune the list of species that we have to consider to a manageable number.

In this section, $\Pi_{i}$ denotes the projection of a triple onto its $i$-th coordinate, and $\Pi_{i,j}$ denotes the projection onto coordinates $i,j$. For every PLS $P$ there is an associated graph formed by taking the triples $\mathcal{T}(P)$ of $P$ as vertices and joining two vertices by an edge if the corresponding triples agree in any coordinate. We say that $P$ is connected if the associated graph is connected, otherwise it is disconnected. For our purposes, we only need to consider connected PLS, because of our next result. For this result only, we need to broaden our definition of embedding to allow the row and column indices of a PLS to be any set of positive integers, rather than insisting that they are consecutive starting at 1. However, this is merely a technicality, which can be overcome by appropriate relabelling.

Lemma 10**.**

Let $P,P_{1},P_{2}$ be PLS for which $\mathcal{T}(P)=T_{1}\cup T_{2}$, where $T_{i}=\mathcal{T}(P_{i})$ for $i=1,2$ and $\Pi_{i}(T_{1})\cap\Pi_{i}(T_{2})=\emptyset$ for $i=1,2,3$. Suppose that there are embeddings $P_{1}\hookrightarrow G$ and $P_{2}\hookrightarrow H$ for groups $G,H$. Then $P$ embeds into $G\times H\times C$, where $C$ is a cyclic group of order $3$.

Proof.

Let $g$ denote a generator for $C$. Suppose that we have embeddings $(\phi_{1},\phi_{2},\phi_{3}):P_{1}\hookrightarrow G$ and $(\phi^{\prime}_{1},\phi^{\prime}_{2},\phi^{\prime}_{3}):P_{2}\hookrightarrow H$. It is routine to verify that the three maps

[TABLE]

are injections that provide an embedding $P\hookrightarrow G\times H\times C$. ∎

In particular, Lemma 10 shows that if $P_{1}$ and $P_{2}$ both embed in finite groups, then so does their disconnected union $P$. Also, if either $P_{1}$ or $P_{2}$ cannot be embedded in an infinite group, then it is immediate that $P$ cannot either. These two observations combine to tell us that the smallest PLS that embeds in an infinite group but not in any finite group will be connected. As an aside, we note that the disconnected PLS in Figure 4 does not embed in the cyclic group of order 6, showing that the cyclic factor $C$ in Lemma 10 cannot be abandoned.

In [13], various restrictions were shown in terms of which PLS need to be considered when studying embeddings in groups. The next lemma is a direct adaptation of [13, Lem.10&11], which helps us to rule out many PLS.

Lemma 11**.**

Let $P$ be a PLS and $T=\{(i,j,P_{i,j}):(i,j)\in\mathcal{S}(P)\}$. Suppose that either

there exists $(r,c,s)\in T$ such that $T^{\prime}=T\setminus\{(r,c,s)\}$ satisfies $r\notin\Pi_{1}(T^{\prime})$ and for each $(r^{\prime},c^{\prime},s^{\prime})\in T^{\prime}$ either $(r^{\prime},c)\in\Pi_{1,2}(T^{\prime})$ or $(r^{\prime},s)\in\Pi_{1,3}(T^{\prime})$, or 2. 2)

there exists $\ell\geqslant 1$ and distinct $(r,c_{1},s_{1}),\dots,(r,c_{\ell},s_{\ell})\in T$ for which $T^{\prime}=T\setminus\{(r,c_{i},s_{i}):1\leqslant i\leqslant\ell\}$ satisfies $r\notin\Pi_{1}(T^{\prime})$ and

[TABLE]

Then the smallest PLS to embed in an infinite group, but not in any finite group, cannot be in the same species as $P$.

For ease of exposition, Lemma 11 has not been stated in a form that displays the symmetry between the three coordinates in $T$. Hence when using it to check whether we need to consider a particular PLS, it is important to consider all three PLS obtained by uniformly cyclically permuting the triples in $T$. We call any connected PLS that cannot be ruled out using Lemma 11 a candidate. Table 1 shows, for each size up to 12, the number of all species of PLS, species of connected PLS, and species of candidate PLS. The catalogue of PLS for each size $s$ was generated by extending each species representative of size $s-1$ in all possible different ways, then using nauty [11] to screen for species representatives. The number of species of PLS for sizes at most $8$ has been independently confirmed by Fálcon and Stones [4]. The number of species of PLS for sizes at most $7$ were previously reported in [13].

6 The outcome

In this section we report the results of our computations, which employed the techniques developed in the previous sections.

For each candidate PLS $P$ as identified in §5, we proceeded as follows. First, we computed $\langle P\rangle$ and a reduced presentation $\langle P\rangle\cong\langle\hat{X}\mid\hat{\mathcal{R}}\rangle$, see §4.1. If we then found duplicates in the list of row labels, column labels, or symbols, then we knew that $P$ could not be embedded in any group, so we stopped considering it. If we found no duplicates, then we proceeded with further tests. If $\hat{\mathcal{R}}=\emptyset$, then $\langle P\rangle$ was a free group and $P$ embedded into a finite group (Corollary 5); we then checked whether $P$ also embedded in a finite abelian group (Lemma 6). Similarly, if $\hat{\mathcal{R}}\neq\emptyset$, then we first checked whether $P$ embedded in a finite abelian group. If not, then we applied Brute Force Techniques I and the Nilpotent Quotients method (§4.3) to find an embedding of $P$ into some finite nonabelian group. If we did not succeed, then we applied the Knuth-Bendix completion algorithm (which might tell us that $P$ cannot be embedded in any group, see §4.1) and the Low Index Subgroup method (which might find an embedding into a finite group, see §4.3). After all these tests, there were only 50 species of PLS remaining for which we did not reach a conclusion. All had size 12. Below we describe how we proved that these PLS can be embedded into an infinite group, but in no finite group. First we summarise in Table 2 the results of our computations for each candidate of size at most $12$. The headings on the columns of Table 2 indicate which family of PLS is being counted, according to the following schema:

[TABLE]

We comment on the 50 PLS of size 12 which can be embedded into an infinite group, but in no finite group. In all but 6 cases, the reduced presentation of $P$ is $\{a,b\mid b=[b,b^{a}]\}$, hence $\langle P\rangle$ is isomorphic to the Baumslag group $G=\langle a,b\mid b=[b,b^{a}]\rangle$ as defined in Lemma 7. In the remaining 6 cases, the reduced presentations show that $\langle P\rangle$ is isomorphic to

[TABLE]

respectively. In Figure 5 we present two PLS $P_{1}$ and $P_{2}$ with corresponding group $B_{1}$ and $B_{2}$, respectively. In $B_{1}$ we have $b^{-1}=[b^{-1},(b^{-2})^{a}]$, and we define $H_{1}=\langle b^{-2},a\rangle$. In $B_{2}$ we observe $[b^{2},(b^{2})^{a}]=[b,(b^{2})^{a}]^{b}[b,(b^{2})^{a}]=b^{b}b=b^{2}$, and we define $H_{2}=\langle b^{2},a\rangle$. It is easy to verify by computer (using coset-enumeration) that $H_{1}=B_{1}$ and $H_{2}=B_{2}$; moreover, the function SearchForIsomorphism provided by the computer algebra system Magma [2] proves that $G\to H_{1}$, $(a,b)\mapsto(a,b^{-2})$, and $G\to H_{2}$, $(a,b)\mapsto(a,b^{2})$, are isomorphisms. In conclusion, $G\cong B_{1}\cong H_{1}\cong B_{2}\cong H_{2}$, and $b^{-2}$ and $b^{2}$ are trivial in every finite quotient of $H_{1}$ and $H_{2}$, respectively.

Each of the 50 PLS embeds into $\langle P\rangle$. This can be proved as in the proof of Lemma 7 by showing that there are no duplicates among the list of symbols, row labels, and columns labels, respectively. In our case, we showed by computer that if there were duplicates, then the resulting group $\langle P\rangle$ was cyclic, which we knew was not the case. This established that each of the 50 PLS embeds into $\langle P\rangle$. In each case, the structure of the embedded PLS implied that no embedding into a finite group exists. For this deduction we made frequent use of the facts that $b=1$ in any finite quotient of $G$ and $b^{2}=1$ in any finite quotient of $H_{1}$ or $H_{2}$. We thereby established Theorem 1.

We conclude with two remarks. First, it is known that the word problem in one-relator groups is solvable, but also that counter-intuitive things can happen. For example, it is shown in [10] that for every integer $t\geqslant 3$, the group $\langle a,b\mid ba^{t}b^{2}a^{t}\rangle$ satisfies $[b^{3},a]=1$, even though the defining relator has length $2t+3$. This indicates that one has to be careful when checking for embeddings into one-relator groups such as the Baumslag group. Words much shorter than the defining relator may, in principle, equate to the identity. Second, recall that Hirsch and Jackson found a PLS of size 29 which can be embedded in an infinite group but in no finite group. As shown in [7, Ex.3.7], their PLS embeds into the group $\langle a,b,c,d\mid ab=b^{2}a,\;bc=c^{2}b,\;cd=d^{2}c,\;da=ad^{2}\rangle$ defined by Higman, who also proved that this group has no nontrivial finite quotients. Thus the example in [7] works for a different underlying reason than the 50 examples mentioned in Theorem 1. Data from this project, including those 50 examples, can be downloaded from [14].