# Quasi-dihedral group

From Encyclopedia of Mathematics

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