Quasidihedral group

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In mathematics, the quasidihedral groups (also known as semidihedral groups) are groups with similar properties to the dihedral groups. In particular they often arise as (somewhat incomplete) symmetry groups of regular polygons, such as the octagon.

For example, in Galois theory the Galois group G of the polynomial

x8 − 2

over the rational field is isomorphic to the quasidihedral group of order 16. In this case generators and relations for G are given by

\langle r,f \mid r^8 = f^2 = 1, frf = r^3\rangle.

This is almost the group generated by a rotation r by an angle π / 4 and a reflection f with axis of symmetry a line through the side of an octagon (that is, the dihedral group D16), except for the fact that frf = r3 instead of r7 so that the relation is with respect to a half rotation (4 - 1 = 3) instead of a full one (8 - 1 = 7).

In general, for every natural number n≥3, there is a quasidihedral group of order 2n, denoted by QD2n. In terms of generators and relations, it can be written as

\langle r, f \mid r^{2^{n - 1}} = f^2 = 1, frf = r^{2^{n - 2}-1}\rangle.

This is clearly a non-abelian 2-group, and in fact the quasidihedral groups appear prominently in the classification of the finite 2-groups.

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