关于自由群的讨论

6.3 A WORD ON FREE GROUPS

6.3 关于自由群的讨论

In this section we introduce the basic theory of so-called free groups. This will enable us to make precise the notions of generators and relations which were used in earlier chapters. The results of this section rely only on the basic theory of homomorphisms.

在这一节中，我们介绍所谓自由群的基本理论。这将使我们能够精确地定义前几章中使用的生成元和关系概念。本节的结果仅依赖于同态的基本理论。

The basic idea of a free group $F\left( S\right)$ generated by a set $S$ is that there are no relations satisfied by any of the elements in $S$ ( $S$ is "free" of relations). For example,if $S$ is the set $\{ a,b\}$ then the elements of the free group on the two generators $a$ and $b$ are of the form $a,{aa},{ab},{abab},{bab},$ etc.,called words in $a$ and $b$ ,together with the inverses of these elements, and all these elements are considered distinct. If we group like terms together,then we obtain elements of the familiar form $a,{b}^{-3},{ab}{a}^{-1}{b}^{2}$ etc. Such elements are multiplied by concatenating their words (for example,the product of $\textit{aba}$ and ${b}^{-1}{a}^{3}b$ would simply be ${aba}{b}^{-1}{a}^{3}b$ ). It is natural at the outset (even before we know $S$ is contained in some group) to simply define $F\left( S\right)$ to be the set of all words in $S,$ where two such expressions are multiplied in $F\left( S\right)$ by concatenating them. Although in essence this is what we do, it is necessary to be more formal in order to prove that this concatenation operation is well defined and associative. After all, even the familiar notation ${a}^{n}$ for the product $a \cdot a\cdots a$ ( $n$ terms) is permissible only because we know that this product is independent of the way it is bracketed (cf. the generalized associative law in Section 1.1). The formal construction of $F\left( S\right)$ is carried out below for an arbitrary set $S$ .

自由群 $F\left( S\right)$ 由集合 $S$ 生成的的基本思想是，集合 $S$ 中的任何元素之间不存在满足的关系（$S$ 是“无关系”的）。例如，如果 $S$ 是集合 $\{ a,b\}$，那么在两个生成元 $a$ 和 $b$ 上的自由群的元素形式为 $a,{aa},{ab},{abab},{bab},$ 等等，称为 $a$ 和 $b$ 中的词，加上这些元素的逆，所有这些元素都被认为是不同的。如果我们把相似的项归在一起，那么我们得到熟悉的形式的元素 $a,{b}^{-3},{ab}{a}^{-1}{b}^{2}$ 等等。这样的元素通过连接它们的词相乘（例如，$\textit{aba}$ 和 ${b}^{-1}{a}^{3}b$ 的乘积将简单地是 ${aba}{b}^{-1}{a}^{3}b$）。在一开始（甚至在知道 $S$ 包含在某些群中之前）就很自然地定义 $F\left( S\right)$ 为所有 $S,$ 中词的集合，其中两个这样的表达式通过连接它们在 $F\left( S\right)$ 中相乘。虽然本质上我们就是这样做的，但是为了证明这种连接运算是良定义且结合的，我们需要更加形式化。毕竟，即使是我们熟悉的乘积 $a \cdot a\cdots a$（$n$ 项）的表示 ${a}^{n}$ 也只有在我们知道这个乘积与括号的方式无关时才是允许的（参见第1.1节中的广义结合律）。对于任意的集合 $S$，$F\left( S\right)$ 的形式构造将在下面进行。

One important property reflecting the fact that there are no relations that must be satisfied by the generators in $S$ is that any map from the set $S$ to a group $G$ can be uniquely extended to a homomorphism from the group $F\left( S\right)$ to $G$ (basically since we have specified where the generators must go and the images of all the other elements are uniquely determined by the homomorphism property - the fact that there are no relations to worry about means that we can specify the images of the generators arbitrarily). This is frequently referred to as the universal property of the free group and in fact characterizes the group $F\left( S\right)$ .

一个反映 $S$ 中不存在必须由生成元满足的关系的重要性质是，从集合 $S$ 到一个群 $G$ 的任何映射都可以唯一地扩展为从群 $F\left( S\right)$ 到 $G$ 的同态（基本原因是我们已经指定了生成元必须去的位置，而所有其他元素的形象都是由同态性质唯一确定的 - 由于没有关系需要担心，我们可以任意指定生成元的形象）。这通常被称为自由群的普遍性质，实际上也表征了群 $F\left( S\right)$ 。

The notion of "freeness" occurs in many algebraic systems and it may already be familiar (using a different terminology) from elementary vector space theory. When the algebraic systems are vector spaces, $F\left( S\right)$ is simply the vector space which has $S$ as a basis. Every vector in this space is a unique linear combination of the elements of $S$ (the analogue of a "word"). Any set map from the basis $S$ to another vector space $V$ extends uniquely to a linear transformation (i.e.,vector space homomorphism) from $F\left( S\right)$ to $V$ .

“自由性”的概念出现在许多代数系统中，并且在初等向量空间理论中可能已经熟悉（使用不同的术语）。当代数系统是向量空间时，$F\left( S\right)$ 只是一个以 $S$ 为基的向量空间。这个空间中的每个向量都是 $S$ 元素的唯一线性组合（"单词"的类似物）。从基 $S$ 到另一个向量空间 $V$ 的任何集合映射都可以唯一地扩展为从 $F\left( S\right)$ 到 $V$ 的线性变换（即，向量空间同态）。

Before beginning the construction of $F\left( S\right)$ we mention that one often sees the universal property described in the language of commutative diagrams. In this form it reads (for groups) as follows: given any set map $\varphi$ from the set $S$ to a group $G$ there is a unique homomorphism $\Phi : F\left( S\right) \rightarrow G$ such that ${\left. \Phi \right| }_{S} = \varphi$ i.e.,such that the following diagram commutes:

在开始构建 $F\left( S\right)$ 之前，我们提一下人们经常看到用交换图的语言描述的普遍性质。在这种形式中，对于群来说，它读作：给定从集合 $S$ 到一个群 $G$ 的任何集合映射 $\varphi$，都存在一个唯一的同态 $\Phi : F\left( S\right) \rightarrow G$ 使得 ${\left. \Phi \right| }_{S} = \varphi$，即，使得以下图表可交换：

As mentioned above,the only difficulty with the construction of $F\left( S\right)$ is the verification that the concatenation operation on the words in $F\left( S\right)$ is well defined and associative. To prove the associative property for multiplication of words we return to the most basic level where all the exponents in the words of $S$ are $\pm 1$ .

如上所述，构建 $F\left( S\right)$ 的唯一困难在于验证 $F\left( S\right)$ 中单词的连接操作是良定义且具有结合性。为了证明单词乘法的结合性质，我们回到了最基础的层面，在那里 $S$ 中所有单词的指数都是 $\pm 1$。

We first introduce inverses for elements of $S$ and an identity.

我们首先为 $S$ 的元素引入逆元和一个单位元。

Let ${S}^{-1}$ be any set disjoint from $S$ such that there is a bijection from $S$ to ${S}^{-1}$ . For each $s \in S$ denote its corresponding element in ${S}^{-1}$ by ${s}^{-1}$ and similarly for each $t \in {S}^{-1}$ let the corresponding element of $S$ be denoted by ${t}^{-1}$ (so ${\left( {s}^{-1}\right) }^{-1} = s$ ). Take a singleton set not contained in $S \cup {S}^{-1}$ and call it {1}. Let ${1}^{-1} = 1$ and for any $x \in S \cup {S}^{-1} \cup \{ 1\}$ let ${x}^{1} = x.$

设 ${S}^{-1}$ 是与 $S$ 不相交的任意集合，并且存在从 $S$ 到 ${S}^{-1}$ 的双射。对于每个 $s \in S$，用 ${s}^{-1}$ 表示其在 ${S}^{-1}$ 中的对应元素，同样对于每个 $t \in {S}^{-1}$，让 $S$ 中的对应元素用 ${t}^{-1}$ 表示（因此 ${\left( {s}^{-1}\right) }^{-1} = s$）。取一个不包含在 $S \cup {S}^{-1}$ 中的单元素集合，称之为 {1}。设 ${1}^{-1} = 1$，对于任何 $x \in S \cup {S}^{-1} \cup \{ 1\}$，设 ${x}^{1} = x.$。

Next we describe the elements of the free group on the set $S$ . A word on $S$ is by definition a sequence

接下来，我们描述在集合 $S$ 上的自由群的元素。根据定义，$S$ 上的单词是一个序列。

(that is,for each sequence there is an $N$ such that ${s}_{i} = 1$ for all $i \geq N$ ). Thus we can think of a word as a finite product of elements of $S$ and their inverses (where repetitions are allowed). Next, in order to assure uniqueness of expressions we consider only words which have no obvious “cancellations” between adjacent terms (such as ${ba}{a}^{-1}b = {bb}$ ). The word $\left( {{s}_{1},{s}_{2},{s}_{3},\ldots }\right)$ is said to be reduced if

（即对于每个序列，都存在一个 $N$，使得 ${s}_{i} = 1$ 对于所有 $i \geq N$）。因此，我们可以将单词视为 $S$ 的元素及其逆元的有限乘积（允许重复）。接下来，为了确保表达式的唯一性，我们只考虑没有明显的“约分”相邻项的单词（例如 ${ba}{a}^{-1}b = {bb}$）。单词 $\left( {{s}_{1},{s}_{2},{s}_{3},\ldots }\right)$ 被称为简化单词，如果。

(1) ${s}_{i + 1} \neq {s}_{i}^{-1}$ for all $i$ with ${s}_{i} \neq 1$ ,and

（1）对于所有 $i$ 且 ${s}_{i} \neq 1$，${s}_{i + 1} \neq {s}_{i}^{-1}$。

(2) if ${s}_{k} = 1$ for some $k$ ,then ${s}_{i} = 1$ for all $i \geq k$ .

（2）如果对于某个 $k$，${s}_{k} = 1$，那么对于所有 $i \geq k$，${s}_{i} = 1$。

The reduced word $\left( {1,1,1,\ldots }\right)$ is called the empty word and is denoted by 1 . We now simplify the notation by writing the reduced word $\left( {{s}_{1}^{{\epsilon }_{1}},{s}_{2}^{{\epsilon }_{2}},\ldots ,{s}_{n}^{{\epsilon }_{n}},1,1,1,\ldots }\right) ,$ ${s}_{i} \in S,{\epsilon }_{i} = \pm 1$ ,as ${s}_{1}^{{\epsilon }_{1}}{s}_{2}^{{\epsilon }_{2}}\ldots {s}_{n}^{{\epsilon }_{n}}$ . Note that by definition,reduced words ${r}_{1}^{{\delta }_{1}}{r}_{2}^{{\delta }_{2}}\ldots {r}_{m}^{{\delta }_{m}}$ and ${s}_{1}^{{\epsilon }_{1}}{s}_{2}^{{\epsilon }_{2}}\ldots {s}_{n}^{{\epsilon }_{n}}$ are equal if and only if $n = m$ and ${\delta }_{i} = {\epsilon }_{i},1 \leq i \leq n$ . Let $F\left( S\right)$ be the set of reduced words on $S$ and embed $S$ into $F\left( S\right)$ by

简化词 $\left( {1,1,1,\ldots }\right)$ 被称为空词，并表示为 1。我们现在简化表示法，将简化词 $\left( {{s}_{1}^{{\epsilon }_{1}},{s}_{2}^{{\epsilon }_{2}},\ldots ,{s}_{n}^{{\epsilon }_{n}},1,1,1,\ldots }\right) ,$ ${s}_{i} \in S,{\epsilon }_{i} = \pm 1$ 写作 ${s}_{1}^{{\epsilon }_{1}}{s}_{2}^{{\epsilon }_{2}}\ldots {s}_{n}^{{\epsilon }_{n}}$。注意，按照定义，简化词 ${r}_{1}^{{\delta }_{1}}{r}_{2}^{{\delta }_{2}}\ldots {r}_{m}^{{\delta }_{m}}$ 和 ${s}_{1}^{{\epsilon }_{1}}{s}_{2}^{{\epsilon }_{2}}\ldots {s}_{n}^{{\epsilon }_{n}}$ 仅当 $n = m$ 和 ${\delta }_{i} = {\epsilon }_{i},1 \leq i \leq n$ 时相等。设 $F\left( S\right)$ 为 $S$ 上的简化词集合，并将 $S$ 嵌入 $F\left( S\right)$ 中通过

Under this set injection we identify $S$ with its image and henceforth consider $S$ as a subset of $F\left( S\right)$ . Note that if $S = \varnothing ,F\left( S\right) = \{ 1\}$ .

在这个集合注入下，我们将 $S$ 与其像识别，并从此将 $S$ 视为 $F\left( S\right)$ 的子集。注意如果 $S = \varnothing ,F\left( S\right) = \{ 1\}$。

We are now in a position to introduce the binary operation on $F\left( S\right)$ . The principal technical difficulty is to ensure that the product of two reduced words is again a reduced word. Although the definition appears to be complicated it is simply the formal rule for "successive cancellation" of juxtaposed terms which are inverses of each other (e.g., $a{b}^{-1}a$ times ${a}^{-1}{ba}$ should reduce to ${aa}$ ). Let ${r}_{1}^{{\delta }_{1}}{r}_{2}^{{\delta }_{2}}\ldots {r}_{m}^{{\delta }_{m}}$ and ${s}_{1}^{{\epsilon }_{1}}{s}_{2}^{{\epsilon }_{2}}\ldots {s}_{n}^{{\epsilon }_{n}}$ be reduced words and assume first that $m \leq n$ . Let $k$ be the smallest integer in the range $1 \leq k \leq m + 1$ such that ${s}_{k}^{{\epsilon }_{k}} \neq {r}_{m - k + 1}^{-{\delta }_{m - k + 1}}.$ Then the product of these reduced words is defined to be:

我们现在可以引入 $F\left( S\right)$ 上的二元运算。主要的技术困难是确保两个简化词的乘积仍然是一个简化词。尽管定义看起来可能很复杂，但它只是“连续消去”相邻的互为逆元的项的正式规则（例如，$a{b}^{-1}a$ 乘以 ${a}^{-1}{ba}$ 应该简化为 ${aa}$）。设 ${r}_{1}^{{\delta }_{1}}{r}_{2}^{{\delta }_{2}}\ldots {r}_{m}^{{\delta }_{m}}$ 和 ${s}_{1}^{{\epsilon }_{1}}{s}_{2}^{{\epsilon }_{2}}\ldots {s}_{n}^{{\epsilon }_{n}}$ 为简化词，并首先假设 $m \leq n$。设 $k$ 为范围 $1 \leq k \leq m + 1$ 中最小的整数，使得 ${s}_{k}^{{\epsilon }_{k}} \neq {r}_{m - k + 1}^{-{\delta }_{m - k + 1}}.$。那么这些简化词的乘积定义为：

The product is defined similarly when $m \geq n$ ,so in either case it results in a reduced word. Theorem 16. $F\left( S\right)$ is a group under the binary operation defined above.

当 $m \geq n$ 时，乘积也以类似方式定义，因此在任何情况下，它都导致一个简化词。定理 16。$F\left( S\right)$ 在上述定义的二元运算下是一个群。

Proof: One easily checks that 1 is an identity and that the inverse of the reduced word ${s}_{1}^{{\epsilon }_{1}}{s}_{2}^{{\epsilon }_{2}}\ldots {s}_{n}^{{\epsilon }_{n}}$ is the reduced word ${s}_{n}^{-{\epsilon }_{n}}{s}_{n - 1}^{-{\epsilon }_{n - 1}}\ldots {s}_{1}^{-{\epsilon }_{1}}$ . The difficult part of the proof is the verification of the associative law. This can be done by induction on the “length” of the words involved and considering various cases or one can proceed as follows: For each $s \in S \cup {S}^{-1} \cup \{ 1\}$ define ${\sigma }_{s} : F\left( S\right) \rightarrow F\left( S\right)$ by

证明：很容易验证 1 是一个单位元，且化简后的单词 ${s}_{1}^{{\epsilon }_{1}}{s}_{2}^{{\epsilon }_{2}}\ldots {s}_{n}^{{\epsilon }_{n}}$ 的逆元是化简后的单词 ${s}_{n}^{-{\epsilon }_{n}}{s}_{n - 1}^{-{\epsilon }_{n - 1}}\ldots {s}_{1}^{-{\epsilon }_{1}}$。证明的困难部分是验证结合律。这可以通过对所涉及单词的“长度”进行归纳，并考虑各种情况来完成，或者也可以如下进行：对于每个 $s \in S \cup {S}^{-1} \cup \{ 1\}$，定义 ${\sigma }_{s} : F\left( S\right) \rightarrow F\left( S\right)$ 通过

Since ${\sigma }_{{s}^{-1}} \circ {\sigma }_{s}$ is the identity map of $F\left( S\right) \rightarrow F\left( S\right) ,{\sigma }_{s}$ is a permutation of $F\left( S\right)$ . Let $A\left( F\right)$ be the subgroup of the symmetric group on the set $F\left( S\right)$ which is generated by $\left\{ {{\sigma }_{s} \mid s \in S}\right\}$ . It is easy to see that the map

由于 ${\sigma }_{{s}^{-1}} \circ {\sigma }_{s}$ 是 $F\left( S\right) \rightarrow F\left( S\right) ,{\sigma }_{s}$ 的恒等映射，$F\left( S\right) \rightarrow F\left( S\right) ,{\sigma }_{s}$ 是 $F\left( S\right)$ 的一个排列。设 $A\left( F\right)$ 是对称群在集合 $F\left( S\right)$ 上的子群，它由 $\left\{ {{\sigma }_{s} \mid s \in S}\right\}$ 生成。很容易看出该映射

is a (set) bijection between $F\left( S\right)$ and $A\left( S\right)$ which respects their binary operations. Since $A\left( S\right)$ is a group,hence associative,so is $F\left( S\right)$ .

是 $F\left( S\right)$ 和 $A\left( S\right)$ 之间的一个（集合）双射，且尊重它们的二元运算。由于 $A\left( S\right)$ 是一个群，因此是结合的，所以 $F\left( S\right)$ 也是结合的。

The universal property of free groups now follows easily.

自由群的普遍性质现在很容易得出。

Theorem 17. Let $G$ be a group, $S$ a set and $\varphi : S \rightarrow G$ a set map. Then there is a unique group homomorphism $\Phi : F\left( S\right) \rightarrow G$ such that the following diagram commutes:

定理 17。设 $G$ 是一个群，$S$ 是一个集合，$\varphi : S \rightarrow G$ 是一个集合映射。那么存在一个唯一的群同态 $\Phi : F\left( S\right) \rightarrow G$，使得以下图表可交换：

Proof: Such a map $\Phi$ must satisfy $\Phi \left( {{s}_{1}^{{\epsilon }_{1}}{s}_{2}^{{\epsilon }_{2}}\ldots {s}_{n}^{{\epsilon }_{n}}}\right) = \varphi {\left( {s}_{1}\right) }^{{\epsilon }_{1}}\varphi {\left( {s}_{2}\right) }^{{\epsilon }_{2}}\ldots \varphi {\left( {s}_{n}\right) }^{{\epsilon }_{n}}$ if it is to be a homomorphism (which proves uniqueness), and it is straightforward to check that this map is in fact a homomorphism (which proves existence).

证明：这样一个映射 $\Phi$ 必须满足 $\Phi \left( {{s}_{1}^{{\epsilon }_{1}}{s}_{2}^{{\epsilon }_{2}}\ldots {s}_{n}^{{\epsilon }_{n}}}\right) = \varphi {\left( {s}_{1}\right) }^{{\epsilon }_{1}}\varphi {\left( {s}_{2}\right) }^{{\epsilon }_{2}}\ldots \varphi {\left( {s}_{n}\right) }^{{\epsilon }_{n}}$，如果它是一个同态（这证明了唯一性），并且直接验证这个映射实际上是一个同态（这证明了存在性）。

Corollary 18. $F\left( S\right)$ is unique up to a unique isomorphism which is the identity map on the set $S$ .

推论 18。$F\left( S\right)$ 在集合 $S$ 上的唯一同构是单位映射。

Proof: This follows from the universal property. Suppose $F\left( S\right)$ and ${F}^{\prime }\left( S\right)$ are two free groups generated by $S$ . Since $S$ is contained in both $F\left( S\right)$ and ${F}^{\prime }\left( S\right)$ ,we have natural injections $S \hookrightarrow {F}^{\prime }\left( S\right)$ and $S \hookrightarrow F\left( S\right)$ . By the universal property in the theorem, it follows that we have unique associated group homomorphisms $\Phi : F\left( S\right) \rightarrow {F}^{\prime }\left( S\right)$ and ${\Phi }^{\prime } : {F}^{\prime }\left( S\right) \rightarrow F\left( S\right)$ which are both the identity on $S$ . The composite ${\Phi }^{\prime }\Phi$ is a homomorphism from $F\left( S\right)$ to $F\left( S\right)$ which is the identity on $S$ ,so by the uniqueness statement in the theorem,it must be the identity map. Similarly $\Phi {\Phi }^{\prime }$ is the identity,so $\Phi$ is an isomorphism (with inverse ${\Phi }^{\prime }$ ),which proves the corollary.

证明：这遵循泛性质。假设 $F\left( S\right)$ 和 ${F}^{\prime }\left( S\right)$ 是由 $S$ 生成的两个自由群。由于 $S$ 同时包含在 $F\left( S\right)$ 和 ${F}^{\prime }\left( S\right)$ 中，我们有两个自然的单射 $S \hookrightarrow {F}^{\prime }\left( S\right)$ 和 $S \hookrightarrow F\left( S\right)$。根据定理中的泛性质，可以得出我们有一个唯一的关联群同态 $\Phi : F\left( S\right) \rightarrow {F}^{\prime }\left( S\right)$ 和 ${\Phi }^{\prime } : {F}^{\prime }\left( S\right) \rightarrow F\left( S\right)$，它们在 $S$ 上都是恒等映射。复合映射 ${\Phi }^{\prime }\Phi$ 是从 $F\left( S\right)$ 到 $F\left( S\right)$ 的同态，且在 $S$ 上是恒等映射，所以根据定理中的唯一性陈述，它必须是恒等映射。类似地，$\Phi {\Phi }^{\prime }$ 也是恒等映射，因此 $\Phi$ 是同构（其逆为 ${\Phi }^{\prime }$），这证明了推论。

Definition. The group $F\left( S\right)$ is called the free group on the set $S$ . A group $F$ is a free group if there is some set $S$ such that $F = F\left( S\right)$ - in this case we call $S$ a set of free generators (or a free basis) of $F$ . The cardinality of $S$ is called the rank of the free group.

定义。群 $F\left( S\right)$ 被称为由集合 $S$ 生成的自由群。如果一个群 $F$ 存在某个集合 $S$ 使得 $F = F\left( S\right)$，在这种情况下我们称 $S$ 为 $F$ 的一组自由生成元（或自由基）。$S$ 的基数被称为自由群的秩。

One can now simplify expressions in a free group by using exponential notation, so we write ${a}^{3}{b}^{-2}$ instead of the formal reduced word ${aaa}{b}^{-1}{b}^{-1}$ . Expressions like ${aba}$ , however,cannot be simplified in the free group on $\{ a,b\}$ . We mention one important theorem in this area.

现在可以使用指数记法来简化自由群中的表达式，因此我们写作 ${a}^{3}{b}^{-2}$ 而不是形式化的简化词 ${aaa}{b}^{-1}{b}^{-1}$。然而，像 ${aba}$ 这样的表达式在 $\{ a,b\}$ 的自由群中不能简化。我们在此领域提及一个重要的定理。

Theorem 19. (Schreier) Subgroups of a free group are free.

定理 19。（施莱尔）自由群的子群是自由的。

This is not trivial to prove and we do not include a proof. There is a nice proof of this result using covering spaces (cf. Trees by J.-P. Serre, Springer-Verlag, 1980).

证明这一点并不简单，我们这里不包含证明。有一个使用覆盖空间来证明这个结果的漂亮证明（参见 J.-P. Serre 的《Trees》，Springer-Verlag，1980）。

Presentations

表示法

Let $G$ be any group. Then $G$ is a homomorphic image of a free group: take $S = G$ and $\varphi$ as the identity map from $G$ to $G$ ; then Theorem 16 produces a (surjective) homomorphism from $F\left( G\right)$ onto $G$ . More generally,if $S$ is any subset of $G$ such that $G = \langle S\rangle$ ,then again there is a unique surjective homomorphism from $F\left( S\right)$ onto $G$ which is the identity on $S$ . (Note that we can now independently formulate the notion that a subset generates a group by noting that $G = \langle S\rangle$ if and only if the map $\pi : F\left( S\right) \rightarrow G$ which extends the identity map of $S$ to $G$ is surjective.)

设 $G$ 为任意群。那么 $G$ 是一个自由群的同态像：取 $S = G$ 和 $\varphi$ 作为从 $G$ 到 $G$ 的恒等映射；那么定理16产生了一个从 $F\left( G\right)$ 到 $G$ 的（满射）同态。更一般地，如果 $S$ 是 $G$ 的任意子集，且 $G = \langle S\rangle$ ，那么同样存在一个从 $F\left( S\right)$ 到 $G$ 的唯一满射同态，它在 $S$ 上是恒等的。（注意，我们现在可以独立地表述一个子集生成一个群的概念，通过注意到 $G = \langle S\rangle$ 当且仅当从 $S$ 的恒等映射扩展到 $G$ 的映射 $\pi : F\left( S\right) \rightarrow G$ 是满射的。）

Definition. Let $S$ be a subset of a group $G$ such that $G = \langle S\rangle$ .

定义。设 $S$ 是群 $G$ 的一个子集，使得 $G = \langle S\rangle$ 。

(1) A presentation for $G$ is a pair(S,R),where $R$ is a set of words in $F\left( S\right)$ such that the normal closure of $\langle R\rangle$ in $F\left( S\right)$ (the smallest normal subgroup containing $\begin{matrix} \langle R\rangle ) & \mathrm{{equals}\;{the}\;{kernel}\;{of}\;{the}\;{homomorphism}} \\ \pi : F\left( S\right) & \rightarrow G\;\mathrm{({where}}\;\pi \;\mathrm{{extends}} \end{matrix}$ the identity map from $S$ to $S$ ). The elements of $S$ are called generators and those of $R$ are called relations of $G$ .

（1）$G$ 的一个表示是一个二元组 (S,R)，其中 $R$ 是 $F\left( S\right)$ 中的一组单词，使得 $\langle R\rangle$ 在 $F\left( S\right)$ 中的正规闭包（包含 $\begin{matrix} \langle R\rangle ) & \mathrm{{equals}\;{the}\;{kernel}\;{of}\;{the}\;{homomorphism}} \\ \pi : F\left( S\right) & \rightarrow G\;\mathrm{({where}}\;\pi \;\mathrm{{extends}} \end{matrix}$ 的最小正规子群，恒等映射从 $S$ 到 $S$）。$S$ 的元素被称为生成元，$R$ 的元素被称为 $G$ 的关系。

(2) We say $G$ is finitely generated if there is a presentation(S,R)such that $S$ is a finite set and we say $G$ is finitely presented if there is a presentation(S,R)with both $S$ and $R$ finite sets.

(2) 我们说 $G$ 是有限生成的，如果存在一个表示 (S,R)，使得 $S$ 是一个有限集，并且我们说 $G$ 是有限表示的，如果存在一个表示 (S,R)，其中 $S$ 和 $R$ 都是有限集。

Note that if(S,R)is a presentation,the kernel of the map $F\left( S\right) \rightarrow G$ is not $\langle R\rangle$ itself but rather the (much larger) group generated by $R$ and all conjugates of elements in $R$ . Note that even for a fixed set $S$ a group will have many different presentations (we can always throw redundant relations into $R$ ,for example). If $G$ is finitely presented with $S = \left\{ {{s}_{1},{s}_{2},\ldots ,{s}_{n}}\right\}$ and $R = \left\{ {{w}_{1},{w}_{2},\ldots ,{w}_{k}}\right\}$ ,we write (as we have in preceding chapters):

注意，如果 (S,R) 是一个表示，那么映射 $F\left( S\right) \rightarrow G$ 的核不是 $\langle R\rangle$ 本身，而是由 $R$ 和 $R$ 中元素的所有共轭生成的（更大的）群。注意，即使对于固定的集 $S$，一个群也会有多种不同的表示（例如，我们总是可以将在 $R$ 中冗余的关系抛出）。如果 $G$ 是由 $S = \left\{ {{s}_{1},{s}_{2},\ldots ,{s}_{n}}\right\}$ 和 $R = \left\{ {{w}_{1},{w}_{2},\ldots ,{w}_{k}}\right\}$ 有限表示的，我们写作（如我们在前几章中所做的那样）：

and if $w$ is the word ${w}_{1}{w}_{2}^{-1}$ ,we shall write ${w}_{1} = {w}_{2}$ instead of $w = 1$ .

如果 $w$ 是单词 ${w}_{1}{w}_{2}^{-1}$，我们将写 ${w}_{1} = {w}_{2}$ 而不是 $w = 1$。

Examples

示例

(1) Every finite group is finitely presented. To see this let $G = \left\{ {{g}_{1},\ldots ,{g}_{n}}\right\}$ be a finite group. Let $S = G$ and let $\pi : F\left( S\right) \rightarrow G$ be the homomorphism extending the identity map of $S$ . Let ${R}_{0}$ be the set of words ${g}_{i}{g}_{j}{g}_{k}^{-1}$ ,where $i,j = 1,\ldots ,n$ and ${g}_{i}{g}_{j} = {g}_{k}$ in G. Clearly ${R}_{0} \leq \ker \pi$ . If $N$ is the normal closure of ${R}_{0}$ in $F\left( S\right)$ and $\widetilde{G} = F\left( S\right) /N$ , then $G$ is a homomorphic image of $\widetilde{G}$ (i.e., $\pi$ factors through $N$ ). Moreover,the set of elements $\left\{ {{\widetilde{g}}_{i} \mid i = 1,\ldots ,n}\right\}$ is closed under multiplication. Since this set generates $\widetilde{G}$ ,it must equal $\widetilde{G}$ . Thus $\left| \widetilde{G}\right| = \left| G\right|$ and so $N = \ker \pi$ and $\left( {S,{R}_{0}}\right)$ is a presentation of $G$ .

(1) 每个有限群都是有限生成的。为了证明这一点，设 $G = \left\{ {{g}_{1},\ldots ,{g}_{n}}\right\}$ 为一个有限群。设 $S = G$ ，并且让 $\pi : F\left( S\right) \rightarrow G$ 是 $S$ 的恒等映射的扩张同态。设 ${R}_{0}$ 是由 ${g}_{i}{g}_{j}{g}_{k}^{-1}$ 组成的单词集合，其中 $i,j = 1,\ldots ,n$ 和 ${g}_{i}{g}_{j} = {g}_{k}$ 属于 G。显然 ${R}_{0} \leq \ker \pi$ 。如果 $N$ 是 ${R}_{0}$ 在 $F\left( S\right)$ 中的正规闭包且 $\widetilde{G} = F\left( S\right) /N$ ，那么 $G$ 是 $\widetilde{G}$ 的同态像（即，$\pi$ 通过 $N$ 分解）。此外，元素集合 $\left\{ {{\widetilde{g}}_{i} \mid i = 1,\ldots ,n}\right\}$ 在乘法下是封闭的。由于这个集合生成 $\widetilde{G}$ ，它必须等于 $\widetilde{G}$ 。因此 $\left| \widetilde{G}\right| = \left| G\right|$ ，所以 $N = \ker \pi$ ，$\left( {S,{R}_{0}}\right)$ 是 $G$ 的一个表示式。

This illustrates a sufficient condition for(S,R)to be a presentation for a given finite group $G$ :

这说明了一个充分条件，使得 (S,R) 对于给定的有限群 $G$ 是一个表示式：

(i) $S$ must be a generating set for $G$ ,and

(i) $S$ 必须是 $G$ 的生成集，并且

(ii) any group generated by $S$ satisfying the relations in $R$ must have order $\leq \left| G\right|$ . (2) Abelian groups can be presented easily. For instance

(ii) 任何由 $S$ 生成的且满足 $R$ 中关系的群必须有阶 $\leq \left| G\right|$ 。(2) 阿贝尔群可以很容易地表示。例如

(Recall $\left\lbrack {a,b}\right\rbrack = {a}^{-1}{b}^{-1}{ab}$ ).

（回顾 $\left\lbrack {a,b}\right\rbrack = {a}^{-1}{b}^{-1}{ab}$ ）。

(3) Some familiar non-abelian groups introduced in earlier chapters have simple presentations:

(3) 在前面的章节中介绍的一些熟悉的非阿贝尔群有简单的表示式：

To check,for example,the presentation for ${D}_{2n}$ note that the relations in the presentation $\langle r,s \mid {r}^{n} = {s}^{2} = 1,{s}^{-1}{rs} = {r}^{-1}\rangle$ imply that this group has a normal subgroup (generated by $r$ ) of order $\leq n$ whose quotient is generated by $s$ (which has order $\leq 2$ ). Thus any group with these generators and relations has order at most ${2n}$ . Since we already know of the existence of the group ${D}_{2n}$ of order ${2n}$ satisfying these conditions, the abstract presentation must equal ${D}_{2n}$ .

例如，要检查 ${D}_{2n}$ 的表示，请注意表示中的关系 $\langle r,s \mid {r}^{n} = {s}^{2} = 1,{s}^{-1}{rs} = {r}^{-1}\rangle$ 暗示这个群有一个阶为 $\leq n$ 的正规子群（由 $r$ 生成），其商由 $s$ 生成（其阶为 $\leq 2$）。因此，任何具有这些生成元和关系的群其阶至多为 ${2n}$。既然我们已经知道存在一个阶为 ${2n}$ 且满足这些条件的群 ${D}_{2n}$，抽象表示必须等于 ${D}_{2n}$。

(4) As mentioned in Section 1.2, in general it is extremely difficult even to determine if a given set of generators and relations is or is not the identity group (let alone determine whether it is some other nontrivial finite group). For example, in the following two presentations the first group is an infinite group and the second is the identity group (cf. Trees, Chapter 1):

（4）如第1.2节所述，通常情况下，甚至确定给定的生成元和关系集合是否为单位群（更不用说确定它是否是其他非平凡有限群）都极其困难。例如，在以下两个表示中，第一个群是无限群，第二个是单位群（参见 Trees，第1章）：

(5) It is easy to see that ${S}_{n}$ is generated by the transpositions (12),(23), $\ldots ,\left( {n - {1n}}\right)$ , and that these satisfy the relations

（5）容易看出 ${S}_{n}$ 由置换（12）、（23）$\ldots ,\left( {n - {1n}}\right)$ 生成，并且这些满足关系

(here $\left| {i - j}\right|$ denotes the absolute value of the integer $i - j$ ). One can prove by induction on $n$ that these form a presentation of ${S}_{n}$ :

（这里 $\left| {i - j}\right|$ 表示整数 $i - j$ 的绝对值）。可以通过对 $n$ 进行归纳来证明这些构成了 ${S}_{n}$ 的表示：

As mentioned in Section 1.6 we can use presentations of a group to find homomorphisms between groups or to find automorphisms of a group. We did this in classifying groups of order 6,for example,when we proved that any non-abelian group of order 6 was generated by an element of order 3 and an element of order 2 inverting it; thus there is a homomorphism from ${S}_{3}$ onto any non-abelian group of order 6 (hence an isomorphism, by computing orders). More generally,suppose $G$ is presented by,say,generators $a,b$ with relations ${r}_{1},\ldots ,{r}_{k}$ . If ${a}^{\prime },{b}^{\prime }$ are any elements of a group $H$ satisfying these relations,there is a homomorphism from $G$ into $H$ . Namely,if $\pi : F\left( {\{ a,b\} }\right) \rightarrow G$ is the presentation homomorphism,we can define ${\pi }^{\prime } : F\left( {\{ a,b\} }\right) \rightarrow H$ by ${\pi }^{\prime }\left( a\right) = {a}^{\prime }$ and ${\pi }^{\prime }\left( b\right) = {b}^{\prime }$ . Then $\ker \pi \leq \ker {\pi }^{\prime }$ so ${\pi }^{\prime }$ factors through $\ker \pi$ and we obtain

如1.6节所述，我们可以使用群的表示来找到群之间的同态或找到群的自同构。例如，在分类阶数为6的群时，我们就这么做了，当时我们证明了任何阶数为6的非阿贝尔群都可以由一个阶数为3的元素和一个阶数为2的元素生成，并对其进行逆变换；因此存在一个从 ${S}_{3}$ 到任何阶数为6的非阿贝尔群的同态（通过计算阶数，因此是一个同构）。更一般地，假设 $G$ 由生成元 $a,b$ 和关系 ${r}_{1},\ldots ,{r}_{k}$ 表示。如果 ${a}^{\prime },{b}^{\prime }$ 是满足这些关系的群 $H$ 中的任意元素，那么存在一个从 $G$ 到 $H$ 的同态。具体来说，如果 $\pi : F\left( {\{ a,b\} }\right) \rightarrow G$ 是表示同态，我们可以通过 ${\pi }^{\prime } : F\left( {\{ a,b\} }\right) \rightarrow H$ 定义 ${\pi }^{\prime }\left( a\right) = {a}^{\prime }$ 和 ${\pi }^{\prime }\left( b\right) = {b}^{\prime }$ 。那么 $\ker \pi \leq \ker {\pi }^{\prime }$ ，因此 ${\pi }^{\prime }$ 可以通过 $\ker \pi$ 分解，我们得到

In,particular,if $\left\langle {{a}^{\prime },{b}^{\prime }}\right\rangle = H = G$ ,this homomorphism is an automorphism of $G$ . Conversely, any automorphism must send a set of generators to another set of generators satisfying the same relations. For example, ${D}_{8} = \left\langle {a,b \mid {a}^{2} = {b}^{4} = 1,{aba} = {b}^{-1}}\right\rangle$ and any pair ${a}^{\prime },{b}^{\prime }$ of elements,where ${a}^{\prime }$ is a noncentral element of order 2 and ${b}^{\prime }$ is of order 4, satisfies the same relations. Since there are four noncentral elements of order 2 and two elements of order 4, ${D}_{8}$ has 8 automorphisms.

特别是，如果 $\left\langle {{a}^{\prime },{b}^{\prime }}\right\rangle = H = G$ ，这个同态是 $G$ 的自同构。反过来，任何自同构都必须将一组生成元映射到满足相同关系的另一组生成元。例如，${D}_{8} = \left\langle {a,b \mid {a}^{2} = {b}^{4} = 1,{aba} = {b}^{-1}}\right\rangle$ 和任意一对元素 ${a}^{\prime },{b}^{\prime }$ ，其中 ${a}^{\prime }$ 是一个非中心阶数为2的元素，而 ${b}^{\prime }$ 的阶数为4，满足相同的关系。因为有四个非中心阶数为2的元素和两个阶数为4的元素，${D}_{8}$ 有8个自同构。

Similarly,any pair of elements of order 4 in ${Q}_{8}$ which are not equal or inverses of each other necessarily generate ${Q}_{8}$ and satisfy the relations given in Example 3 above. It is easy to check that there are 24 such pairs, so

同样，${Q}_{8}$ 中任何阶为4的元素对，如果不相等也不是彼此的逆元，必然生成 ${Q}_{8}$ 并满足上述例3中给出的关系。容易验证存在24对这样的元素，所以

Free objects can be constructed in (many, but not all) other categories. For instance, a monoid is a set together with a binary operation satisfying all of the group axioms except the axiom specifying the existence of inverses. Free objects in the category of monoids play a fundamental role in theoretical computer science where they model the behavior of machines (Turing machines, etc.). We shall encounter free algebras (i.e., polynomial algebras) and free modules in later chapters.

在（许多但并非所有）其他类别中可以构造自由对象。例如，一个幺半群是一个带有满足所有群公理（除了指定存在逆元的公理之外）的二进制运算的集合。在幺半群类别中的自由对象在理论计算机科学中扮演了基本角色，它们模拟了机器（图灵机等）的行为。我们将在后续章节遇到自由代数（即多项式代数）和自由模块。

EXERCISES

练习

1. Let ${F}_{1}$ and ${F}_{2}$ be free groups of finite rank. Prove that ${F}_{1} \cong {F}_{2}$ if and only if they have the same rank. What facts do you need in order to extend your proof to infinite ranks (where the result is also true)?

2. 设 ${F}_{1}$ 和 ${F}_{2}$ 是有限秩的自由群。证明 ${F}_{1} \cong {F}_{2}$ 当且仅当它们具有相同的秩。你需要哪些事实来将你的证明扩展到无限秩（在此结果同样成立）？

3. Prove that if $\left| S\right| > 1$ then $F\left( S\right)$ is non-abelian.

4. 证明如果 $\left| S\right| > 1$ ，那么 $F\left( S\right)$ 是非交换的。

5. Prove that the commutator subgroup of the free group on 2 generators is not finitely generated (in particular, subgroups of finitely generated groups need not be finitely generated).

6. 证明2生成元的自由群的换位子群不是有限生成的（特别地，有限生成群的子群不一定是有限生成的）。

7. Prove that every nonidentity element of a free group is of infinite order.

8. 证明自由群中每个非单位元素都是无限阶的。

9. Establish a finite presentation for ${A}_{4}$ using 2 generators.

10. 用2个生成元为 ${A}_{4}$ 建立一个有限呈现。

11. Establish a finite presentation for ${S}_{4}$ using 2 generators.

12. 用2个生成元为 ${S}_{4}$ 建立一个有限呈现。

13. Prove that the following is a presentation for the quaternion group of order 8 :

14. 证明以下是对阶为8的四元数群的呈现：

8. Use presentations to find the orders of the automorphism groups of the groups ${Z}_{2} \times {Z}_{4}$ and ${Z}_{4} \times {Z}_{4}$ .

9. 使用呈现来找到群 ${Z}_{2} \times {Z}_{4}$ 和 ${Z}_{4} \times {Z}_{4}$ 的自同构群的阶。

10. Prove that $\operatorname{Aut}\left( {Q}_{8}\right) \cong {S}_{4}$ .

11. 证明 $\operatorname{Aut}\left( {Q}_{8}\right) \cong {S}_{4}$ 。

12. This exercise exhibits an automorphism of ${S}_{6}$ that is not inner (hence,together with Exercise 19 in Section 4.4 it shows that $\left| {\operatorname{Aut}\left( {S}_{6}\right) : \operatorname{Inn}\left( {S}_{6}\right) }\right| = 2$ ). Let ${t}_{1}^{\prime } = \left( {1\;2}\right) \left( {3\;4}\right) \left( {5\;6}\right)$ , ${t}_{2}^{\prime } = \left( {1\;4}\right) \left( {2\;5}\right) \left( {3\;6}\right) ,{t}_{3}^{\prime } = \left( {1\;3}\right) \left( {2\;4}\right) \left( {5\;6}\right) ,{t}_{4}^{\prime } = \left( {1\;2}\right) \left( {3\;6}\right) \left( {4\;5}\right) ,\mathrm{{and}}\;{t}_{5}^{\prime } = \left( {1\;4}\right) \left( {2\;3}\right) \left( {5\;6}\right) .$ Show that ${t}_{1}^{\prime },\ldots ,{t}_{5}^{\prime }$ satisfy the following relations:

13. 这个练习展示了 ${S}_{6}$ 的一个自同构，它不是内自同构（因此，与第4.4节中的练习19一起，它表明了 $\left| {\operatorname{Aut}\left( {S}_{6}\right) : \operatorname{Inn}\left( {S}_{6}\right) }\right| = 2$ ）。设 ${t}_{1}^{\prime } = \left( {1\;2}\right) \left( {3\;4}\right) \left( {5\;6}\right)$ ，${t}_{2}^{\prime } = \left( {1\;4}\right) \left( {2\;5}\right) \left( {3\;6}\right) ,{t}_{3}^{\prime } = \left( {1\;3}\right) \left( {2\;4}\right) \left( {5\;6}\right) ,{t}_{4}^{\prime } = \left( {1\;2}\right) \left( {3\;6}\right) \left( {4\;5}\right) ,\mathrm{{and}}\;{t}_{5}^{\prime } = \left( {1\;4}\right) \left( {2\;3}\right) \left( {5\;6}\right) .$ 证明 ${t}_{1}^{\prime },\ldots ,{t}_{5}^{\prime }$ 满足以下关系：

${\left( {t}_{i}^{\prime }\right) }^{2} = 1$ for all $i$ ,

${\left( {t}_{i}^{\prime }\right) }^{2} = 1$ 对所有 $i$ 成立，

${\left( {t}_{i}^{\prime }{t}_{j}^{\prime }\right) }^{2} = 1$ for all $i$ and $j$ with $\left| {i - j}\right| \geq 2$ ,and

${\left( {t}_{i}^{\prime }{t}_{j}^{\prime }\right) }^{2} = 1$ 对所有 $i$ 和 $j$ 成立，其中 $\left| {i - j}\right| \geq 2$ ，

${\left( {t}_{i}^{\prime }{t}_{i + 1}^{\prime }\right) }^{3} = 1$ for all $i \in \{ 1,2,3,4\}$ .

${\left( {t}_{i}^{\prime }{t}_{i + 1}^{\prime }\right) }^{3} = 1$ 对所有 $i \in \{ 1,2,3,4\}$ 成立。

Deduce that ${S}_{6} = \left\langle {{t}_{1}^{\prime },\ldots ,{t}_{5}^{\prime }}\right\rangle$ and that the map

推导出 ${S}_{6} = \left\langle {{t}_{1}^{\prime },\ldots ,{t}_{5}^{\prime }}\right\rangle$ 并且映射

extends to an automorphism of ${S}_{6}$ (which is clearly not inner since it does not send transpositions to transpositions). [Use the presentation for ${S}_{6}$ described in Example 5.]

扩展为 ${S}_{6}$ 的自同构（这显然不是内自同构，因为它不将置换映射到置换）。[使用示例5中描述的 ${S}_{6}$ 的表示法。]

11. Let $S$ be a set. The group with presentation(S,R),where $R = \{ \left\lbrack {s,t}\right\rbrack \mid s,t \in S\}$ is called the free abelian group on $S$ -denote it by $A\left( S\right)$ . Prove that $A\left( S\right)$ has the following universal property: if $G$ is any abelian group and $\varphi : S \rightarrow G$ is any set map,then there is a unique group homomorphism $\Phi : A\left( S\right) \rightarrow G$ such that ${\left. \Phi \right| }_{S} = \varphi$ . Deduce that if $A$ is a free abelian group on a set of cardinality $n$ then

12. 设 $S$ 为一个集合。以 (S,R) 为生成元和关系的群被称为在 $S$ 上的自由阿贝尔群，记作 $A\left( S\right)$ 。证明 $A\left( S\right)$ 具有以下普遍性质：如果 $G$ 是任意阿贝尔群，$\varphi : S \rightarrow G$ 是任意集合映射，那么存在唯一的群同态 $\Phi : A\left( S\right) \rightarrow G$ 使得 ${\left. \Phi \right| }_{S} = \varphi$ 。推导出如果 $A$ 是在基数 $n$ 的集合上的自由阿贝尔群，那么

12. Let $S$ be a set and let $c$ be a positive integer. Formulate the notion of a free nilpotent group on $S$ of nilpotence class $c$ and prove it has the appropriate universal property with respect to nilpotent groups of class $\leq c$ .

13. 设 $S$ 为一个集合，$c$ 为一个正整数。构建在 $S$ 上的 nilpotence 类 $c$ 的自由 nilpotent 群的概念，并证明它具有关于类 $\leq c$ 的 nilpotent 群的适当普遍性质。

14. Prove that there cannot be a nilpotent group $N$ generated by two elements with the property that every nilpotent group which is generated by two elements is a homomorphic image of $N$ (i.e.,the specification of the class $c$ in the preceding problem was necessary).

15. 证明不存在由两个元素生成的幂零群 $N$，使得每一个由两个元素生成的幂零群都是 $N$ 的同态像（即，前一个问题中对类 $c$ 的指定是必要的）。