Theory of Groups of Finite Order

by W. Burnside

Overview

After introducing permutation notation and defining group, the author discusses the simpler properties of group that are independent of their modes of representation; composition-series of groups; isomorphism of a group with itself; Abelian groups; groups whose orders are the powers of primes; Sylow's theorem; more. 18 illustrations. A classic introduction.

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ISBN-13:	9780486159447
Publisher:	Dover Publications
Publication date:	01/23/2013
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	544
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Theory of Groups of Finite Order

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By W. Burnside

Dover Publications, Inc.

Copyright © 2004 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-15944-7

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CHAPTER 1

ON PERMUTATIONS.

1. AMONG 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 a1, a2, ..., an 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 distinct letters are replaced by one and the same letter, is called a permutation performed on the n letters. Such a permutation will change any given arrangement

a1, a2,...., an

of the n letters into a definite new arrangement

b1, b2, ..., bn

of the same n letters.

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

(a1, a2, ..., an b1, b2, ..., bn),

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 permutation; by the following process, the expression of the permutation 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' in the upper line, which has not yet occurred, and let q', r', ... follow it as q, r,... followed p, till we arrive at s' in the upper line with p' standing under it. If the set of n letters is still not exhausted, repeat the process, starting with a letter p' 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that the permutation 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

(pqr ... s)

the operation of replacing p by q, q by r, ..., and s by p, the permutation will be completely represented by the symbol

(pqr ...s) (p'q'r' ... s') (p"q"r" ... s")....

The advantage of this mode of expressing the permutation is that each of the letters occurs only once in the symbol.

4. The separate components of the above symbol, such as (pqr ... s), are called the cycles of the permutation. 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 permutation. The brackets enclosing single letters may clearly be omitted without risk of ambiguity, as also may the unaltered letters themselves. Thus the permutation

(a, b, c, d, e c, b, d, a, e)

may be written (acd) (b) (e), or (acd) be, or simply (acd). If for any reason it were desirable to indicate that permutations 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 permutation is not unique. The symbol (qr ... sp) clearly represents the same permutation as (pqr ... s), if the letters that occur between r and s in the two symbols are the same and occur in the same sequence; 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 permutation; 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 permutations successively, it is seen at once that the order in which the permutations are carried out in general affects the result. Thus to give a very simple instance, the permutation (ab) followed by (ac) changes a into b, since b is unaltered by the second permutation. Again, (ab) changes b into a and (ac) changes a into c, so that the two permutations performed successively change b into c. Lastly, (ab) does not affect c and (ac) changesc into a. Hence the two permutations performed successively change a into b, b into c, c into a, and affect no other symbols. The result of the two permutations performed successively is therefore equivalent to the permutation (abc); and it may be similarly shewn that (ac) followed by (ab) gives (acb) as the resulting permutation. To avoid ambiguity it is therefore necessary to assign, once for all, the meaning to be attached to such a symbol as s1s2, where s1 and s2 are the symbols of two given permutations. We shall always understand by the symbol s1s2the result of carrying out first the permutation s1and then the permutation s2. Thus the two simple examples given above may be expressed in the form

(ab)(ac) = (abc), (ac)(ab) = (acb),

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

If now

s1s2 = s4 and s2s3 = s5,

the symbol s1s2s3 may be regarded as the permutation s4 followed by s3 or as s1followed by s5. But if s1 changes any letter a into b, while s2 changes b into c and s3changes c into d, then s4 changes a into c and s5 changes b into d. Hence s4s3 and s1s] ITL5both change a into d; and therefore, a being any letter operated upon by the permutations,

s4s3 = s1s5.

Hence the meaning of the symbol s1s2s3 is definite; it depends only on the component permutations s1, s2, s3 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 symbols representing the successive performance of any number of permutations. To avoid circumlocution, it is convenient to speak of the permutation s1s3 ... sn as the product of the permutations s1, s2, ..., sn in the sequence given. The product of a number of permutations, thus defined, always obeys the associative law but does not in general obey the commutative law of algebraical multiplication.

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

s = (a1, a2, ..., an b1, b2, ..., bn),

then

s-1 = (b1, b2, ..., bn a1, a2, ..., an).

Let s0 denote the identical permutation which can be represented by

(a1, a2, ..., an a1, a2, ..., an).

Then

ss-1 = s0 and s-1s = s0,

so that s is the permutation inverse to s-1.

Now if

ts = t's,

then

tss-1 = t'ss-1,

or

ts0 = t's0.

But ts0 is the same permutation as t, since s0 produces no change; and therefore

t = t'.

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

st = st'

involves

t = t'.

8. The result of performing r times in succession the same permutation s is represented symbolically by sr. Since, as has been seen, products of permutations obey the associative law of multiplication, it follows that

sμ sv = sμ+v = svsμ.

Now since there are only a finite number of distinct permutations that can be performed on a given finite set of symbols, the series of permutations s, s2, s3, ... cannot be all distinct. Suppose that sm+n is the first of the series which is the same as one that precedes it, and let that one be sn. Then

sm+n = sn,

and therefore

smsn(sn)-1 = sn(sn)-1,

or

sm = s0.

Hence n must be 1. Moreover there is no index μ smaller than m for which this relation holds. For if

sμ = s0,

then

sμ + 1 = ss0 = s,

contrary to the supposition that sm+1 is the first of the series which is the same as s.

Moreover the m-1 permutations s, s2, ..., sm+1 must be all distinct. For if

sμ = sv, v < μ < m,

then

sμ-vsv(sv)-1 = sv(sv)-1,

or

sμ-v = s0,

which has just been shewn to be impossible.

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

sn = s0,

it may be shewn at once that n is a multiple of m the order of 8; and secondly, if

sα = sβ,

then

α - β [equivalent to] 0 (mod. m).

If now the equation

sμ+v = sμsv

be assumed to hold, when either or both of the integers μ and v is a negative integer, a definite meaning is obtained for the symbol s-r, implying the negative power of a permutation; and a definite meaning is also obtained for s0. For

sμs-v = sμ-vsv(sv)-1 = sμ(sv)-1,

so that

s-v = (sv)-1.

Similarly it can be shewn that

s0 = s0.

9. If the cycles of a permutation

s= (pqr ... s) (p 'q' ... s') (p" q" ... s") ...

contain m, m', m", ... letters respectively, and if

sμ = s0,

μ must be a common multiple of m, m', m",.... For sμ changes p into a letter μ places from it in the cyclical set p, q, r,..., s; and therefore, if it changes p into itself, μ must be a multiple of m. In the same way, it must be a multiple of m', m",.... Hence the order of 8 is the least common multiple of m, m', m",....

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

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

10. Two permutations, which contain the same number of cycles and the same number of letters in corresponding cycles, are called similar. If s, s' are similar permutations, so also clearly are sr, s'r; and the orders of s and s' are the same.

Let now

s = (apaq ... a8) (ap'aq' ... a8') ...

and

t = (a1, a2, ..., an) b1, b2, ..., bn)

be any two permutations. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the latter form of the permutation being obtained by actually carrying out the component permutations of the earlier form. Hence s and t-1st are similar permutations.

(Continues...)

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Excerpted from Theory of Groups of Finite Order by W. Burnside. Copyright © 2004 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents

Partial contents:
On Permutation
The Definition of a Group
On the Simpler Properties of a Group Which Are Independent of Its Mode of Representation
Composition-Series of Groups
Isomorphism of a Group with Itself
Abelian Groups
Groups Whose Orders are the Powers of Primes
Sylow's Theorem
Permutation Groups
Groups of Linear Substitutions
Graphical Representation of Groups
Congruence Groups