Theory of Groups of Finite Order

William S. Burnside

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486159447
• Topic: Mathematics
• Subtopic: Group Theory
• Index: Mathematics

An unabridged republication of the classic 1911 edition, this volume constitutes both a great historical contribution tomathematical literature and a basic reference book in its field. Suitable for advanced undergraduates and graduate students in mathematics as well as historians of mathematics, the introductory treatment was hailed by International Mathematical News as "more easily comprehensible than most other books on the subject, " and as "a classic work, extraordinarily rich, " by Elemente der Mathematik.
After introducing permutation notation and presenting the definition of a group, author William Burnside discusses the simpler properties of groups 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; and Sylow's theorem. Permutation groups and groups of linear substitutions receive an extensive treatment; two chapters are devoted to the graphic representation of groups, and the closing chapter examines congruence groups. Forty-five pages of notes at the back of the book offer ample treatment of special topics.

CHAPTER I.

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,..., 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 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,...., a_n

of the n letters into a definite new arrangement

b₁, b₂,..., b_n

of the same n letters.

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

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

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 p...