Theorem 14.1. Let G be a group and a,b,cG.

  1. If ab=ac then b=c. (left cancellation law)
  2. If ba=ca then b=c. (right cancellation law)
  3. ax=b has a unique solution in G, x=a1b.
    xa=b has unique solutions in G, x=ba1.
  4. (a1)1=a.
  5. (ab)1=b1a1.

Note. In the Cayley table of G, ax will be in the row labeled by a.

Note. If a,xG, a finite group, Theorem 14.1 implies ax appears in the row labeled by a in G's Cayley table.

Note. If bG, a finite group, and ax=b, Theorem 14.1(c) implies b appears exactly once in the row labeled by a in G's Cayley table.

Theorem. Ignoring row labels, each element of a finite group appears exactly once in each row of the Cayley table for the group.

Theorem. Ignoring column labels, each element of a finite group appears exactly once in each column of the Cayley table for the group.

Definition. Define integral powers of aG: a0=ea1=aa2=aaan+1=ana so that an equals the product of n a's for each positive integer n. Also, for each positive integer n, an=(a1)n

Theorem. Laws of exponents. For all integers m,n, aman=am+n(am)n=amn

Note. In additive notation, for all integers m,n: na=a++a(n terms)(n)a=n(a)(ma)+(na)=(m+n)an(ma)=(mn)a

Definition. If G is a group and aG then a will denote the set of all integral powers of a. a={annZ}

Example 14.1. The set of all integral powers of (1 2 3) is (1 2 3)={(1),(1 2 3),(1 3 2)}

Theorem 14.2. If G is a group and aG then a, the set of all integral powers of a, is a subgroup of G. The subgroup a, is called the subgroup generated by a.

Definition. If H is a subgroup and H=a, for some aH, then H is said to be a cyclic subgroup.

Guevara Corollary. If aG, then the subgroup a is cyclic.

Example. The group of integers is cyclic: Z=1=1

Theorem 14.3. If G is a group, aG, and r,sZ such that rs and ar=as, then

  1. There is a smallest positive integer n such that an=e.
  2. If t is an integer, then at=e iff n is a divisor of t.
  3. The elements e=a0, a, a2, , an1 are distinct, and a={e,a,a2,,an1}.

Definition. If a is an element of a group, then the smallest positive integer n such that an=e, if it exists, is called the order of a. If there is no such integer, then a is said to have infinite order. The order of an element a will be denoted by o(a).

Note. If aG then an=e automatically if n=0. By the preceding definition, the order of a is never 0 That is, if n is the order of a, then n>0 by definition, and is the smallest such n that an=e.

Example 14.2.

  1. In S3, o((1 2 3))=3.
  2. In the group of nonzero rational numbers (operation multiplication), 2 has infinite order, because 2n1 for every positive integer n.

Note. In additive notation, the condition an=e becomes na=0. In Zn, the condition an=e becomes n[a]=[0].

Example 14.3. In Z6, o([2])=3, because [2][0] and 2[2]=[2][2]=[4]0 but 3[2]=[2][2][2]=[6]=[0]. (We also see that [2]={[0],[2],[4]}.)

Corollary. If a is an element of a group, then o(a)=|a|.