14 ELEMENTARY PROPERTIES OF GROUPS

Theorem 14.1. Let $G$ be a group and $a,b,c\in G.$

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 $G,$ $x=a^{-1}b.$
   $xa=b$ has unique solutions in $G,$ $x=ba^{-1}.$
4. $(a^{-1})^{-1}=a.$
5. $(ab)^{-1}=b^{-1}a^{-1}.$

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

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

Note. If $b\in G,$ 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 $a\in G:$ $$\begin{align*} a^0 &=e\\ a^1 &=a\\ a^2 &=aa\\ &\;\;\vdots\\ a^{n+1} &=a^na \end{align*}$$ so that $a^n$ equals the product of $n$ $a$'s for each positive integer $n.$ Also, for each positive integer $n,$ $$a^{-n}=(a^{-1})^n$$

Theorem. Laws of exponents. For all integers $m,n,$ $$\begin{align*} a^ma^n=a^{m+n}\\ (a^m)^n=a^{mn}\\ \end{align*}$$

Note. In additive notation, for all integers $m,n:$ $$\begin{align*} na &=a+\cdots+a\quad(n\text{ terms})\\ (-n)a &=n(-a)\\ (ma)+(na) &=(m+n)a\\ n(ma) &=(mn)a \end{align*}$$

Definition. If $G$ is a group and $a\in G$ then $\gen{a}$ will denote the set of all integral powers of $a.$ $$\gen{a}=\{a^n\mid n\in\Z\}$$

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

Theorem 14.2. If $G$ is a group and $a\in G$ then $\gen{a},$ the set of all integral powers of $a,$ is a subgroup of $G.$ The subgroup $\gen{a},$ is called the subgroup generated by $a.$

Definition. If $H$ is a subgroup and $H=\gen{a},$ for some $a\in H,$ then $H$ is said to be a cyclic subgroup.

Guevara Corollary. If $a\in G,$ then the subgroup $\gen{a}$ is cyclic.

Example. The group of integers is cyclic: $$\Z=\gen{1}=\gen{-1}$$

Theorem 14.3. If $G$ is a group, $a\in G,$ and $r,s\in\Z$ such that $r\ne s$ and $a^r=a^s,$ then

1. There is a smallest positive integer $n$ such that $a^n=e.$
2. If $t$ is an integer, then $a^t=e$ iff $n$ is a divisor of $t.$
3. The elements $e=a^0,$ $a,$ $a^2,$ $\ldots,$ $a^{n-1}$ are distinct, and $\gen{a}=\{e,a,a^2,\ldots,a^{n-1}\}.$

Definition. If $a$ is an element of a group, then the smallest positive integer $n$ such that $a^n=e,$ 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 $a\in G$ then $a^n=e$ 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 $n\gt0$ by definition, and is the smallest such $n$ that $a^n=e.$

Example 14.2.

1. In $S_3,$ $o((1\ 2\ 3 ))=3.$
2. In the group of nonzero rational numbers (operation multiplication), $2$ has infinite order, because $2^n\ne1$ for every positive integer $n.$

Note. In additive notation, the condition $a^n=e$ becomes $na=0.$ In $\Z_n,$ the condition $a^n=e$ becomes $n[a]=[0].$

Example 14.3. In $\Z_6,$ $o([2])=3,$ because $[2]\ne[0]$ and $$2[2]=[2]\oplus[2]=[4]\ne0$$ but $$3[2]=[2]\oplus[2]\oplus[2]=[6]=[0].$$ (We also see that $\langle[2]\rangle=\{[0],[2],[4]\}.$)

Corollary. If $a$ is an element of a group, then $o(a)=\abs{\gen{a}}.$