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Order of an element

We now set on to investigate the elements and properties of general groups.

$\begin{de} Let $a$\ be an element of a group $G$. The {\em order} of $a$\ is the... ...$n$such that $a^n=e$\ if there is such a number, or infinite otherwise. \end{de}$

$\begin{exs} Every reflection has order 2. Rotation $\rho$\ in the dihedral group... ...e) order. The identity is the only element of order $1$\ in any group. \end{exs}$

$\begin{ex} The orders of the elements of ${\mathbb{Z}}_{10}$\ (under addition modulo $10$) are 1, 10, 5, 10, 5, 2, 5, 10, 5, 10 respectively. \end{ex}$

$\begin{ex} Consider the permutation $ \alpha=(1\ 2)(3\ 4\ 5) $from $S_5$. Its po... ...\ &&\alpha^6=\id. \end{eqnarray*}Hence, the order of $\alpha$\ is $6$. \end{ex}$

$\begin{exc} Find the orders of the elements of the multiplicative group ${\mathbb{Z}}_{7}\setminus\{0\}$. \end{exc}$

$\begin{exc} Find the orders of elements of the dihedral group $D_6$. \end{exc}$

$\begin{thm} If $G$\ is a finite group, then every element of $G$\ has finite order. \end{thm}$

$\begin{proof} Let $a\in G$\ be arbitrary. Consider the elements $a, a^2, a^3,\ld... ... the smallest such positive integer, which is then the order of $a$. \end{proof}$

$\begin{thm} Let $G$\ be a group, and let $a$\ be an element of order $n$\ in $G$... ... if $n\vert(i-j)$. (In particular $a^i=e$\ if and only if $n\vert i$.) \end{thm}$

$\begin{proof} ($\Leftarrow$) If $n\vert(i-j)$\ then write $i-j=qn$, so that \beg... ... of $a$\ and $r<n$, we conclude that $r=0$, and hence $n\vert(i-j)$. \end{proof}$

We are now going to see how to find the orders of elements in various specific groups introduced earlier.

$\begin{thm} The order of an element $a\in {\mathbb{Z}}_n$\ (under addition modulo $n$) is $n/\gcd(a,n)$. \end{thm}$

$\begin{proof} Let the order of $a$\ be $m$; this means that, modulo $n$, we have... ... \end{displaymath}We conclude that $m=n_1=n/\gcd(a,n)$, as required. \end{proof}$

$\begin{thm} The order of an $l$-cycle $\gamma=(i_1\ i_2\ \ldots\ i_l)$\ in the s... ...e $\gamma_i$\ has length $l_i$, is equal to $q=\lcm (l_1,\ldots,l_m)$. \end{thm}$

$\begin{proof} % latex2html id marker 1968Clearly $\gamma^l=\id$. For $j<l$\ we... ...ows that $l_j\vert r$\ for all $j=1,\ldots ,m$, and hence $q\leq r$. \end{proof}$

$\begin{exc} List the elements of $S_3$\ and their orders. \end{exc}$

Edmund F Robertson

11 September 2006