Page:Popular Science Monthly Volume 64.djvu/375

Rh any one of these permutations is repeated, or is followed by some other permutation in the totality, the result is equivalent to a single permutation in the totality. This property is characteristic; for if any set of distinct permutations possesses this property they form a permutation group and it is possible to construct an infinite number of expressions such that they are unchanged by these permutations but by no others.

Soon after the fundamental properties of permutation groups became known, it was observed that many other operations possess the same properties. This gradually led to more abstract definitions of the term group. According to the earliest of these any set of distinct operations such that no additional operation is obtained by repeating one of them or combining any two of them was called a group. All the later definitions included this property, but they generally add other conditions. These additional conditions are frequently satisfied by the nature of the operations which are under consideration and hence do not always require attention. This may account for the fact that the oldest definition is still very commonly met in text-books, notwithstanding the fact that the ablest writers on the subject abandoned it a long time ago.

The three additional conditions which a set of distinct operations must satisfy in order that it becomes a group when the operations are combined are: (1) The associative law must be satisfied; i. e., if $$r, s, t$$ represent any three operations of the set, then the three successive operations $$rst$$ must give the same result independently of the fact whether we replace $$rs$$ or $$st$$ by a single operation. The operations are, however, not generally commutative, that is, $$rs$$ may be different from $$sr$$. (2) From each of the two equations $$rs=ts$$, $$sr=st$$ it follows that $$r = t$$. (3) If the equation $$xy = z$$ involves two operations of the set the third element of the equation must also represent an operation in the set. It may be observed that the totality of integers combined by multiplication obey all these conditions except the last. Hence this totality does not form a group with respect to multiplication, although the contrary has frequently been affirmed.

One of the simplest instances of a group of operations is furnished by the $$n$$ different numbers which satisfy the equation $$ x^n=1 $$. It is very easy to see that these numbers obey each of the four given conditions when they are combined by multiplication. Hence we say that the $$n$$ roots of the equation $$ x^n=1 $$ form a group with respect to multiplication. Since all these roots are powers of a single one of them