Quasidihedral group

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In mathematics, the quasi-dihedral groups and semi-dihedral groups are non-abelian groups of order a power of 2. For every positive integer n greater than or equal to 4, there are exactly four isomorphism classes of nonabelian groups of order 2n which have a cyclic subgroup of index 2. Two are well established, the generalized quaternion group and the dihedral group, but the other two have conflicting names, so this article discusses both groups.

One of the groups is often considered particularly important, since it is an example of a 2-group of maximal nilpotency class. In Bertram Huppert's influential text Endliche Gruppen, this group is called a Quasidiedergruppe. In Daniel Gorenstein's influential text, Finite Groups, this group is called the semidihedral group. Dummit and Foote refer to it as the quasidihedral group; we adopt that name in this article. All give the same presentation for this group:

\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs = r^{2^{n-2}-1}\rangle\,\!.

The other non-abelian 2-group with cyclic subgroup of index 2 is not given a special name in either text, but referred to as just G or M_m(2). In this article, this group is called the other-maximal-cyclic group. Dummit and Foote refer to this group as the modular group. Its presentation in both texts is:

\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs = r^{2^{n-2}+1}\rangle\,\!.

Both these two groups and the dihedral group are semidirect products of a cyclic group <r> of order 2n-1 with a cyclic group <s> of order 2. Such a non-abelian semi-direct product is uniquely determined by an element of order 2 in the group of units of the ring \Bbb{Z}/2^{n-1}\Bbb{Z} and there are precisely three such elements, 2^{n-1}-1, 2^{n-2}-1, and 2^{n-2}+1, corresponding to the dihedral group, the quasidihedral, and the other-maximal-cyclic group.

The generalized quaternion group, the dihedral group, and the quasidihedral group of order 2n all have nilpotency class n−1, and are the only isomorphism classes of groups of order 2n with nilpotency class n−1. The groups of order pn and nilpotency class n−1 were the beginning of the classification of all p-groups via coclass. The other-maximal-cyclic group of order 2n always has nilpotency class 2. This makes the other-maximal-cyclic group less interesting, since most groups of order pn for large n have nilpotency class 2 and have proven difficult to understand directly.

The generalized quaternion, the dihedral, and the quasidihedral group are the only 2-groups whose derived subgroup has index 4. The Alperin–Brauer–Gorenstein theorem classifies the simple groups, and to a degree the finite groups, with quasidihedral Sylow 2-subgroups.