Semidihedral group: Difference between revisions

Revision as of 02:10, 21 August 2011

Definition

Let $n$ be a natural number greater than or equal to $4$ . The semidihedral group or quasidihedral group of order $2^{n}$ (and degree $2^{n - 1}$ ), denoted $S D_{2^{n}}$ , is defined by the following presentation:

$S D_{2^{n}} = Q D_{2^{n}} : = ⟨ a, x ∣ a^{2^{n - 1}} = x^{2} = e, x a x = a^{2^{n - 2} + 1} ⟩$

(here, $e$ is the symbol for the identity element).

Particular cases

$n$	$2^{n}$	$n - 1$	$2^{n - 1}$	Group
4	16	3	8	semidihedral group:SD16
5	32	4	16	semidihedral group:SD32
6	64	5	32	semidihedral group:SD64
7	128	6	64	semidihedral group:SD128

@@ Line 1: / Line 1: @@
 ==Definition==
-Let <math>n</math> be a [[natural number]] greater than or equal to <math>4</math>. The '''semidihedral group''' of order <math>2^n</math> (and '''degree''' <math>2^{n-1}</math>), denoted <math>SD_{2^n}</math>, is defined by the following [[presentation of a group|presentation]]:
+Let <math>n</math> be a [[natural number]] greater than or equal to <math>4</math>. The '''semidihedral group''' or '''quasidihedral group''' of order <math>2^n</math> (and '''degree''' <math>2^{n-1}</math>), denoted <math>SD_{2^n}</math>, is defined by the following [[presentation of a group|presentation]]:
-<math>\! SD_{2^n} := \langle a,x \mid a^{2^{n-1}} = x^2 = e, xax = a ^{2^{n-2} + 1} \rangle</math>
+<math>\! SD_{2^n} = QD_{2^n} := \langle a,x \mid a^{2^{n-1}} = x^2 = e, xax = a ^{2^{n-2} + 1} \rangle</math>
 (here, <math>e</math> is the symbol for the identity element).