Quasi-dihedral group

In mathematics, quasi-dihedral groups are certain finite non-Abelian groups of the order , where is. ${\displaystyle 2^{n}}$ ${\displaystyle n\geq 4}$

definition

A quasi-dihedral group is a group made up of two elements and of form ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle \langle a,b\mid a^{2^{n-1}}=b^{2}=1,bab=a^{2^{n-2}-1}\rangle }$

is generated with. ${\displaystyle n\geq 4}$

Number of elements

It follows because of that . So every finite product of the generators and the quasi-dihedral group can be brought into the form by applying this rule . Because of : ${\displaystyle bab=a^{2^{n-2}-1}}$ ${\displaystyle b^{2}=1}$ ${\displaystyle ba=a^{2^{n-2}-1}b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a^{i}b^{j}}$ ${\displaystyle a^{2^{n-1}}=b^{2}=1}$

The quasi-dihedral group has 2 ⁿ elements: ${\displaystyle \{1,a,a^{2},\ldots ,a^{2^{n-1}},b,ba,ba^{2},\ldots ,ba^{2^{n-1}}\}}$

example

The smallest quasi-dihedral group has the order and is generated by two elements and which satisfy the equations and . Since , after right multiplication with that , it follows from the last equation . So you can in any sequence of 's and ' s each in front of a standing behind the take when this by replacing. It then follows that all elements of this group are of the form . Furthermore, all multiplications in the group can be determined with the above equations. As an example we consider the two products from and : ${\displaystyle 16}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a^{8}=b^{2}=1}$ ${\displaystyle bab=a^{3}}$ ${\displaystyle b^{2}=1}$ ${\displaystyle b}$ ${\displaystyle ba=a^{3}b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle a^{3}}$ ${\displaystyle 1,a,a^{2},\ldots ,a^{7},b,ab,\ldots ,a^{7}b}$ ${\displaystyle a^{2}}$ ${\displaystyle a^{3}b}$

${\displaystyle a^{2}\cdot a^{3}b=a^{5}b}$     (because ) ${\displaystyle a^{2}a^{3}=a^{5}}$

${\displaystyle a^{3}b\cdot a^{2}=a^{3}a^{3}ba=a^{3}a^{3}a^{3}b=a^{9}b=ab}$     ( bring to the right twice and use) ${\displaystyle b}$ ${\displaystyle a^{8}=1}$

Overall, we get the following linkage table

${\displaystyle \,\cdot }$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$
${\displaystyle \,1}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$
${\displaystyle \,a}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$
${\displaystyle \,a^{2}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$	${\displaystyle \,ab}$
${\displaystyle \,a^{3}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$
${\displaystyle \,a^{4}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$
${\displaystyle \,a^{5}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$
${\displaystyle \,a^{6}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$
${\displaystyle \,a^{7}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$
${\displaystyle \,b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$
${\displaystyle \,ab}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$
${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$
${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$
${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$
${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$
${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$
${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$

See also

• Dihedral group
• List of small groups

literature

• Bertram Huppert : Finite groups (= The basic teachings of the mathematical sciences in individual representations. Vol. 134, ISSN  0072-7830 ). Volume 1. Springer, Berlin et al. 1967, pp. 90-93.