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Difference between revisions of "Quasi-dihedral group"

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A finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076450/q0764501.png" />-group defined by generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076450/q0764502.png" /> and defining relations
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A finite $2$-group defined by generators $x,y$ and defining relations
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076450/q0764503.png" /></td> </tr></table>
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$$x^{2^{m-1}}=y^2=x^{-1+2^{m-2}}yx^{-1}y=1,$$
  
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076450/q0764504.png" />. The order of a quasi-dihedral group is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076450/q0764505.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076450/q0764506.png" />. A quasi-dihedral group is sometimes called a semi-dihedral group.
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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====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR>
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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=18888
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
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