In mathematics, quasi-dihedral groups are certain finite non-Abelian groups of the order , where is. 2 n {\ displaystyle 2 ^ {n}} n ≥ 4th {\ displaystyle n \ geq 4}
A quasi-dihedral group is a group made up of two elements and of form a {\ displaystyle a} b {\ displaystyle b}
is generated with. n ≥ 4th {\ displaystyle n \ geq 4}
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 : b a b = a 2 n - 2 - 1 {\ displaystyle bab = a ^ {2 ^ {n-2} -1}} b 2 = 1 {\ displaystyle b ^ {2} = 1} b a = a 2 n - 2 - 1 b {\ displaystyle ba = a ^ {2 ^ {n-2} -1} b} a {\ displaystyle a} b {\ displaystyle b} a i b j {\ displaystyle a ^ {i} b ^ {j}} a 2 n - 1 = b 2 = 1 {\ displaystyle a ^ {2 ^ {n-1}} = b ^ {2} = 1}
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 : 16 {\ displaystyle 16} a {\ displaystyle a} b {\ displaystyle b} a 8th = b 2 = 1 {\ displaystyle a ^ {8} = b ^ {2} = 1} b a b = a 3 {\ displaystyle bab = a ^ {3}} b 2 = 1 {\ displaystyle b ^ {2} = 1} b {\ displaystyle b} b a = a 3 b {\ displaystyle ba = a ^ {3} b} a {\ displaystyle a} b {\ displaystyle b} a {\ displaystyle a} b {\ displaystyle b} a {\ displaystyle a} a 3 {\ displaystyle a ^ {3}} 1 , a , a 2 , ... , a 7th , b , a b , ... , a 7th b {\ displaystyle 1, a, a ^ {2}, \ ldots, a ^ {7}, b, ab, \ ldots, a ^ {7} b} a 2 {\ displaystyle a ^ {2}} a 3 b {\ displaystyle a ^ {3} b}
Overall, we get the following linkage table