Standard

## Chapter I.On Substitutions.

A mong the various notations used in the following pages, there is one of such frequent recurrence that a certain readiness in its use is very desirable in dealing with the subject of this treatise. We therefore propose to devote a preliminary chapter to explaining it in some detail.

2. Let ${a}_{1}$, ${a}_{2}$, …, ${a}_{n}$ be a set of $n$ distinct letters. The operation of replacing each letter of the set by another, which may be the same letter or a diﬀerent one, when carried out under the condition that no two letters are replaced by one and the same letter, is called a substitution performed on the $n$ letters. Such a substitution will change any given arrangement

${a}_{1},\phantom{\rule{1em}{0ex}}{a}_{2},\phantom{\rule{1em}{0ex}}\dots ,\phantom{\rule{1em}{0ex}}{a}_{n}$
of the $n$ letters into a deﬁnite new arrangement
${b}_{1},\phantom{\rule{1em}{0ex}}{b}_{2},\phantom{\rule{1em}{0ex}}\dots ,\phantom{\rule{1em}{0ex}}{b}_{n}$
of the same $n$ letters.

3. One obvious form in which to write the substitution is

$\left(\genfrac{}{}{0.0pt}{}{{a}_{1},{a}_{2},\dots ,{a}_{n}}{{b}_{1},{b}_{2},\dots ,{b}_{n}}\right)$
thereby indicating that each letter in the upper line is to be replaced by the letter standing under it in the lower. The disadvantage of this form is its unnecessary complexity, each of the $n$ letters occurring twice in the expression for the substitution; by the following process, the expression of the substitution may be materially simpliﬁed.

Let $p$ be any one of the $n$ letters, and $q$ the letter in the lower line standing under $p$ in the upper. Suppose now that $r$ is the letter in the lower line that stands under $q$ in the upper, and so on. Since the number of letters is ﬁnite, we must arrive at last at a letter $s$ in the upper line under which $p$ stands. If the set of $n$ letters is not thus exhausted, take any letter $p\prime$ in the upper line, which has not yet occurred, and let $q\prime$, $r\prime$, … follow it as $q$, $r$, … followed $p$, till we arrive at $s\prime$ in the upper line with $p\prime$ standing under it. If the set of $n$ letters is still not exhausted, repeat the process, starting with a letter $p\prime \prime$ which has not yet occurred. Since the number of letters is ﬁnite, we must in this way at last exhaust them; and the $n$ letters are thus distributed into a number of sets

$\begin{array}{ccccc}p,\hfill & q,\hfill & r,\hfill & \dots ,\hfill & s;\hfill \\ p\prime ,\hfill & q\prime ,\hfill & r\prime ,\hfill & \dots ,\hfill & s\prime ;\hfill \\ p\prime \prime ,\hfill & q\prime \prime ,\hfill & r\prime \prime ,\hfill & \dots ,\hfill & s\prime \prime ;\hfill \\ \multicolumn{5}{c}{....................;}\\ \hfill \end{array}$
such that the substitution replaces each letter of a set by the one following it in that set, the last letter of each set being replaced by the ﬁrst of the same set.

If now we represent by the symbol

the operation of replacing $p$ by $q$, $q$ by $r$, …, and $s$ by $p$, the substitution will be completely represented by the symbol
The advantage of this mode of expressing the substitution is that each of the letters occurs only once in the symbol.

4. The separate components of the above symbol, such as are called the cycles of the substitution. In particular cases, one or more of the cycles may contain a single letter; when this happens, the letters so occurring singly are unaltered by the substitution. The brackets enclosing single letters may clearly be omitted without risk of ambiguity, as also may the unaltered letters themselves. Thus the substitution

$\left(\genfrac{}{}{0.0pt}{}{a,b,c,d,e}{c,b,d,a,e}\right)$
may be written , or , or simply . If for any reason it were desirable to indicate that substitutions of the ﬁve letters $a$, $b$, $c$, $d$, $e$ were under consideration, the second of these three forms would be used.

5. The form thus obtained for a substitution is not unique. The symbol clearly represents the same substitution as , if the letters that occur between $r$ and $s$ in the two symbols are the same and occur in the same order; so that, as regards the letters inside the bracket, any one may be chosen to stand ﬁrst so long as the cyclical order is preserved unchanged.

Moreover the order in which the brackets are arranged is clearly immaterial, since the operation denoted by any one bracket has no eﬀect on the letters contained in the other brackets. This latter property is characteristic of the particular expression that has been obtained for a substitution; it depends upon the fact that the expression contains each of the letters once only.

6. When we proceed to consider the eﬀect of performing two or more substitutions successively, it is seen at once that the order in which the substitutions are carried out in general aﬀects the result. Thus to give a very simple instance, the substitution  followed by  changes $a$ into $b$, since $b$ is unaltered by the second substitution. Again,  changes $b$ into $a$ and  changes $a$ into $c$, so that the two substitutions performed successively change $b$ into $c$. Lastly, does not aﬀect $c$ and  changes $c$ into $a$. Hence the two substitutions performed successively change $a$ into $b$, $b$ into $c$, $c$ into $a$, and aﬀect no other symbols. The result of the two substitutions performed successively is therefore equivalent to the substitution ; and it may be similarly shewn that  followed by gives  as the resulting substitution. To avoid ambiguity it is therefore necessary to assign, once for all, the meaning to be attached to such a symbol as ${s}_{1}{s}_{2}$, where ${s}_{1}$ and ${s}_{2}$ are the symbols of two given substitutions. We shall always understand by the symbol ${s}_{1}{s}_{2}$ the result of carrying out ﬁrst the substitution ${s}_{1}$ and then the substitution ${s}_{2}$. Thus the two simple examples given above may be expressed in the form

the sign of equality being used to represent that the substitutions are equivalent to each other.

If now

the symbol ${s}_{1}{s}_{2}{s}_{3}$ may be regarded as the substitution ${s}_{4}$ followed by ${s}_{3}$ or as ${s}_{1}$ followed by ${s}_{5}$. But if ${s}_{1}$ changes any letter $a$ into $b$, while ${s}_{2}$ changes $b$ into $c$ and ${s}_{3}$ changes $c$ into $d$, then ${s}_{4}$ changes $a$ into $c$ and ${s}_{5}$ changes $b$ into $d$. Hence ${s}_{4}{s}_{3}$ and ${s}_{1}{s}_{5}$ both change $a$ into $d$; and therefore, $a$ being any letter operated upon by the substitutions,
${s}_{4}{s}_{3}={s}_{1}{s}_{5}.$

Hence the meaning of the symbol ${s}_{1}{s}_{2}{s}_{3}$ is deﬁnite; it depends only on the component substitutions ${s}_{1}$, ${s}_{2}$, ${s}_{3}$ and their sequence, and it is independent of the way in which they are associated when their sequence is assigned. And the same clearly holds for the symbol representing the successive performance of any number of substitutions. To avoid circumlocution, it is convenient to speak of the substitution ${s}_{1}{s}_{2}\dots {s}_{n}$ as the product of the substitutions ${s}_{1}$, ${s}_{2}$, …, ${s}_{n}$ in the sequence given. The product of a number of substitutions, thus deﬁned, always obeys the associative law but does not in general obey the commutative law of algebraical multiplication.

7. The substitution which replaces every symbol by itself is called the identical substitution. The inverse of a given substitution is that substitution which, when performed after the given substitution, gives as result the identical substitution. Let ${s}_{-1}$ be the substitution inverse to $s$, so that, if

$s=\left(\genfrac{}{}{0.0pt}{}{{a}_{1},{a}_{2},\dots ,{a}_{n}}{{b}_{1},{b}_{2},\dots ,{b}_{n}}\right),$
then
${s}_{-1}=\left(\genfrac{}{}{0.0pt}{}{{b}_{1},{b}_{2},\dots ,{b}_{n}}{{a}_{1},{a}_{2},\dots ,{a}_{n}}\right).$
Let ${s}_{0}$ denote the identical substitution which can be represented by
$\left(\genfrac{}{}{0.0pt}{}{{a}_{1},{a}_{2},\dots ,{a}_{n}}{{a}_{1},{a}_{2},\dots ,{a}_{n}}\right).$

Then

so that $s$ is the substitution inverse to ${s}_{-1}$.

Now if

$ts=t\prime s,$
then
$ts{s}_{-1}=t\prime s{s}_{-1},$
or
$t{s}_{0}=t\prime {s}_{0}.$

But $t{s}_{0}$ is the same substitution as $t$, since ${s}_{0}$ produces no change; and therefore

$t=t\prime .$

In exactly the same way, it may be shewn that the relation

$st=st\prime$
involves
$t=t\prime .$

8. The result of performing $r$ times in succession the same substitution $s$ is represented symbolically by ${s}^{r}$. Since, as has been seen, products of substitutions obey the associative law of multiplication, it follows that

${s}^{\mu }{s}^{\nu }={s}^{\mu +\nu }={s}^{\nu }{s}^{\mu }.$

Now since there are only a ﬁnite number of distinct substitutions that can be performed on a given ﬁnite set of symbols, the series of substitutions $s$, ${s}^{2}$, ${s}^{3}$, … cannot be all distinct. Suppose that ${s}^{m+1}$ is the ﬁrst of the series which is the same as $s$, so that

${s}^{m+1}=s.$

Then

${s}^{m}s{s}_{-1}=s{s}_{-1},$
or
${s}^{m}={s}_{0}.$

There is no index $\mu$ smaller than $m$ for which this relation holds. For if

${s}^{\mu }={s}_{0},$
then
${s}^{\mu +1}=s{s}_{0}=s,$
contrary to the supposition that ${s}^{m+1}$ is the ﬁrst of the series which is the same as $s$.

Moreover the $m-1$ substitutions $s$, ${s}^{2}$, …, ${s}^{m-1}$ must be all distinct. For if

${s}^{\mu }={s}^{\nu },\phantom{\rule{1em}{0ex}}\nu <\mu
then
or
${s}^{\mu -\nu }={s}_{0},$
which has just been shewn to be impossible.

The number $m$ is called the order of the substitution $s$. In connection with the order of a substitution, two properties are to be noted. First, if

${s}^{n}={s}_{0},$
it may be shewn at once that $n$ is a multiple of $m$ the order of $s$; and secondly, if
${s}^{\alpha }={s}^{\beta },$
then

If now the equation

${s}^{\mu +\nu }={s}^{\mu }{s}^{\nu }$
be assumed to hold, when either or both of the integers $\mu$ and $\nu$ is a negative integer, a deﬁnite meaning is obtained for the symbol ${s}^{-r}$, implying the negative power of a substitution; and a deﬁnite meaning is also obtained for ${s}^{0}$. For
so that
Similarly it can be shewn that
${s}^{0}={s}_{0}.$

Since every power of ${s}_{0}$ is the same as ${s}_{0}$, and since wherever ${s}_{0}$ occurs in the symbol ${s}_{1}{s}_{2}\dots {s}_{n}$ of a compound substitution it may be omitted without aﬀecting the result, it is clear that no ambiguity will result from replacing ${s}_{0}$ everywhere by $1$; in other words, we may use $1$ to represent the identical substitution which leaves every letter unchanged. But when this is done, it must of course be remembered that the equation

${s}^{m}=1$
is not a reducible algebraical equation, which is capable of being written in the form

Indeed the symbol $s+s\prime$, where $s$ and $s\prime$ are any two substitutions, has no meaning.

9. If the cycles of a substitution

contain $m$, $m\prime$, $m\prime \prime$, … letters respectively, and if
${s}^{\mu }=1,$
$\mu$ must be a common multiple of $m$, $m\prime$, $m\prime \prime$, …. For ${s}^{\mu }$ changes $p$ into a letter $\mu$ places from it in the cyclical set $p$$q$, $r$, …, $s$; and therefore, if it changes $p$ into itself, $\mu$ must be a multiple of $m$. In the same way, it must be a multiple of $m\prime$, $m\prime \prime$, …. Hence the order of $s$ is the least common multiple of $m$, $m\prime$, $m\prime \prime$, ….

In particular, when a substitution consists of a single cycle, its order is equal to the number of letters which it interchanges. Such a substitution is called a circular substitution.

A substitution, all of whose cycles contain the same number of letters, is said to be regular in the letters which it interchanges; the order of such a substitution is clearly equal to the number of letters in one of its cycles.

10. Two substitutions, which contain the same number of cycles and the same number of letters in corresponding cycles, are called similar. If $s$$s\prime$ are similar substitutions, so also clearly are ${s}^{r}$$s{\prime }^{r}$; and the orders of $s$ and $s\prime$ are the same.

Let now

and
$t=\left(\genfrac{}{}{0.0pt}{}{{a}_{1},{a}_{2},\dots ,{a}_{n}}{{b}_{1},{b}_{2},\dots ,{b}_{n}}\right)$
be any two substitutions. Then

the latter form of the substitution being obtained by actually carrying out the component substitutions of the earlier form. Hence $s$ and ${t}^{-1}st$ are similar substitutions.

Since

${s}_{2}{s}_{1}={s}_{1}^{-1}{s}_{1}{s}_{2}{s}_{1},$
it follows that ${s}_{1}{s}_{2}$ and ${s}_{2}{s}_{1}$ are similar substitutions and therefore that they are of the same order. Similarly it may be shewn that ${s}_{1}{s}_{2}{s}_{3}\dots {s}_{n}$, ${s}_{2}{s}_{3}\dots {s}_{n}{s}_{1}$, …, ${s}_{n}{s}_{1}\dots {s}_{2}{s}_{3}$ are all similar substitutions.

It may happen in particular cases that $s$ and ${t}^{-1}st$ are the same substitution. When this is so, $t$ and $s$ are permutable, that is, $st$ and $ts$ are equivalent to one another; for if

$s={t}^{-1}st,$
then
$ts=st.$

This will certainly be the case when none of the symbols that are interchanged by $t$ are altered by $s$; but it may happen when $s$ and $t$ operate on the same symbols. Thus if

then

Ex. 1. Shew that every regular substitution is some power of a circular substitution.

Ex. 2. If $s$$s\prime$ are permutable regular substitutions of the same $mn$ letters of orders $m$ and $n$, these numbers being relatively prime, shew that $ss\prime$ is a circular substitution in the $mn$ letters.

Ex. 33. If

shew that $s$ is permutable with both ${s}_{1}$ and ${s}_{2}$, and that it can be formed by a combination of ${s}_{1}$ and ${s}_{2}$.

Ex. 4. Shew that the only substitutions of $n$ given letters which are permutable with a circular substitution of the $n$ letters are the powers of the circular substitution.

Ex. 5. Determine all the substitutions of the ten symbols involved in

which are permutable with $s$.

The determination of all the substitutions which are permutable with a given substitution will form the subject of investigation in Chapter X.

11. A circular substitution of order two is called a transposition. It may be easily veriﬁed that

so that every circular substitution can be represented as a product of transpositions; and thence, since every substitution is the product of a number of circular substitutions, every substitution can be represented as a product of transpositions. It must be remembered, however, that, in general, when a substitution is represented in this way, some of the letters will occur more than once in the symbol, so that the order in which the constituent transpositions occur is essential. There is thus a fundamental diﬀerence from the case when the symbol of a substitution is the product of circular substitutions, no two of which contain a common letter.

Since

every transposition, and therefore every substitution of $n$ letters, can be expressed in terms of the $n-1$ transpositions

The number of diﬀerent ways in which a given substitution may be represented as a product of transpositions is evidently unlimited; but it may be shewn that, however the representation is eﬀected, the number of transpositions is either always even or always odd. To prove this, it is suﬃcient to consider the eﬀect of a transposition on the square root of the discriminant of the $n$ letters, which may be written

The transposition changes the sign of the factor ${a}_{r}-{a}_{s}$. When $q$ is less than either $r$ or $s$, the transposition interchanges the factors ${a}_{q}-{a}_{r}$ and ${a}_{q}-{a}_{s}$; and when $q$ is greater than either $r$ or $s$, it interchanges the factors ${a}_{r}-{a}_{q}$ and ${a}_{s}-{a}_{q}$. When $q$ lies between $r$ and $s$, the pair of factors ${a}_{r}-{a}_{q}$ and ${a}_{q}-{a}_{s}$ are interchanged and are both changed in sign. Hence the eﬀect of the single transposition on $D$ is to change its sign. Since any substitution can be expressed as the product of a number of transpositions, the eﬀect of any substitution on $D$ must be either to leave it unaltered or to change its sign. If a substitution leaves $D$ unaltered it must, when expressed as a product of transpositions in any way, contain an even number of transpositions; and if it changes the sign of $D$, every representation of it, as a product of transpositions, must contain an odd number of transpositions. Hence no substitution is capable of being expressed both by an even and by an odd number of transpositions.

A substitution is spoken of as odd or even, according as the transpositions which enter into its representation are odd or even in number.

Further, an even substitution can always be represented as a product of circular substitutions of order three. For any even substitution of $n$ letters can be represented as the product of an even number of the $n-1$ transpositions

in appropriate sequence and with the proper number of occurrences; and the product of any consecutive pair of these is the circular substitution .

Now

so that every circular substitution of order three displacing ${a}_{1}$, and therefore every even substitution of $n$ letters, can be expressed in terms of the $n-2$ substitutions

and their powers.

Ex. 1. Shew that every even substitution of $n$ letters can be expressed in terms of

when $n$ is odd; and in terms of
when $n$ is even.

Ex. 2. If $n+1$ is odd, shew that every even substitution of $mn+1$ letters can be expressed in terms of

and if $n+1$ is even, that every substitution of $mn+1$ letters can be expressed in terms of this set of $m$ circular substitutions.

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