Namespaces
Variants
Actions

Difference between revisions of "Quasi-dihedral group"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX)
m (link)
 
Line 4: Line 4:
 
$$x^{2^{m-1}}=y^2=x^{-1+2^{m-2}}yx^{-1}y=1,$$
 
$$x^{2^{m-1}}=y^2=x^{-1+2^{m-2}}yx^{-1}y=1,$$
  
where $m\geq4$. The order of a quasi-dihedral group is $2^m$; the group has a cyclic invariant subgroup of index 2. The name was given because of the similarity of the defining relations with those of a [[Dihedron group|dihedron group]]; however, a quasi-dihedral group is not isomorphic to the latter for any value of $m$. A quasi-dihedral group is sometimes called a semi-dihedral group.
+
where $m\geq4$. The order of a quasi-dihedral group is $2^m$; the group has a cyclic invariant subgroup of index 2. The name was given because of the similarity of the defining relations with those of a [[dihedral group]]; however, a quasi-dihedral group is not isomorphic to the latter for any value of $m$. A quasi-dihedral group is sometimes called a semi-dihedral group.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR>
 +
</table>

Latest revision as of 09:07, 19 October 2014

A finite $2$-group defined by generators $x,y$ and defining relations

$$x^{2^{m-1}}=y^2=x^{-1+2^{m-2}}yx^{-1}y=1,$$

where $m\geq4$. The order of a quasi-dihedral group is $2^m$; the group has a cyclic invariant subgroup of index 2. The name was given because of the similarity of the defining relations with those of a dihedral group; however, a quasi-dihedral group is not isomorphic to the latter for any value of $m$. A quasi-dihedral group is sometimes called a semi-dihedral group.

References

[1] B. Huppert, "Endliche Gruppen" , 1 , Springer (1967)
How to Cite This Entry:
Quasi-dihedral group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-dihedral_group&oldid=33926
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article