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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