Difference between revisions of "Quasi-dihedral group"
From Encyclopedia of Mathematics
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$$x^{2^{m-1}}=y^2=x^{-1+2^{m-2}}yx^{-1}y=1,$$ | $$x^{2^{m-1}}=y^2=x^{-1+2^{m-2}}yx^{-1}y=1,$$ | ||
| − | 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 [[ | + | 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
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