Order of a group element

In the mathematical sub-area of group theory , the order of a group element or element order of an element of a group is understood to be the smallest natural number for which applies, where is the neutral element of the group. There is no such number, it is said, have infinite order. Elements of finite order are also called torsion elements. The order is sometimes referred to as or . ${\displaystyle g}$ ${\displaystyle (G,\cdot )}$ ${\displaystyle n>0}$ ${\displaystyle g^{n}=e}$ ${\displaystyle e}$ ${\displaystyle g}$ ${\displaystyle \operatorname {ord} (g)}$ ${\displaystyle \operatorname {o} (g)}$

The power of a group element is inductively defined for natural exponents : ${\displaystyle g^{n}}$ ${\displaystyle g}$ ${\displaystyle n\geq 0}$

• ${\displaystyle g^{0}:=e}$
• ${\displaystyle g^{k+1}:=g^{k}\cdot g}$ for all natural ${\displaystyle k\geq 0}$

If the number is finite, it is called the group exponent . ${\displaystyle \exp(G):=\operatorname {kgV} \left\{\operatorname {ord} (g)\,|\,g\in G\right\}}$

properties

• According to Lagrange's theorem , all elements of a finite group have a finite order, and this is a factor of the group order , i. H. the number of elements in the group.
• Conversely, in a finite group, according to Cauchy's theorem, for every prime divisor of the group order there is an element that has the order . No general statement is possible for compound factors (while the neutral element belongs to the trivial factor 1 ). ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle e=e^{1}}$
• The order of an element is equal to the order of the subgroup , of this element generates is.
• It applies if and only if is a multiple of the order of the element . ${\displaystyle g^{d}=e}$ ${\displaystyle d}$ ${\displaystyle \operatorname {ord} (g)}$ ${\displaystyle g}$
• In Abelian groups , the order of the product is a factor of the least common multiple of the orders of and . No such statement is possible in non-Abelian groups; for example, the element of group SL ₂ ( Z ) has infinite order, although it is the product of the elements of order 4 and of order 6. ${\displaystyle g\cdot h}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle \left[\!{\begin{smallmatrix}1&1\\0&1\end{smallmatrix}}\!\right]}$ ${\displaystyle \left[\!{\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}}\!\right]}$ ${\displaystyle \left[\!{\begin{smallmatrix}0&-1\\1&1\end{smallmatrix}}\!\right]}$

literature

• JC Jantzen, J. Schwermer: Algebra. Springer, Berlin / Heidelberg 2006, ISBN 3-540-21380-5 .