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In [[mathematics]], the '''quasi-dihedral groups''' and '''semi-dihedral groups''' are non-abelian [[group (mathematics)|group]]s 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 2<sup>n</sup> which have a cyclic subgroup of [[Index of a subgroup|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 class|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:


:<math>\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs = r^{2^{n-2}-1}\rangle\,\!</math>.
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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:
 
:<math>\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs = r^{2^{n-2}+1}\rangle\,\!</math>.
 
Both these two groups and the dihedral group are [[semidirect product]]s of a cyclic group <''r''> of order 2<sup>n-1</sup> 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 <math>\Bbb{Z}/2^{n-1}\Bbb{Z}</math> and there are precisely three such elements, <math>2^{n-1}-1</math>, <math>2^{n-2}-1</math>, and <math>2^{n-2}+1</math>, 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 2<sup>''n''</sup> all have nilpotency class ''n''&minus;1, and are the only isomorphism classes of groups of order 2<sup>''n''</sup> with nilpotency class ''n''&minus;1. The groups of order ''p''<sup>''n''</sup> and nilpotency class ''n''&minus;1 were the beginning of the classification of all [[p-group]]s via [[coclass (group theory)|coclass]]. The other-maximal-cyclic group of order 2<sup>n</sup> always has nilpotency class 2.  This makes the other-maximal-cyclic group less interesting, since most groups of order ''p''<sup>''n''</sup> for large ''n'' have nilpotency class 2 and have proven difficult to understand directly. <!-- I need to go get a copy of this folklore statement. the result should be in {{MR|113948}} or {{MR|169921}} -->
 
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.
 
==References==
*{{cite book
| last1 = Dummit
| first1 = D. S.
| last2 = Foote
| first2 = R.
| title = Abstract Algebra
| publisher = Wiley
| pages = 71-72
| year = 2004
| edition = 3
| isbn = 9780471433347
}}
* {{cite book
| last = Huppert
| first = B.
| authorlink = Bertram Huppert
| title = Endliche Gruppen
| id = {{MathSciNet | id = 0224703}}
| year = 1967
| publisher = Springer
| pages = 90–93
}}
* {{cite book
| last = Gorenstein
| first = D.
| authorlink = Daniel Gorenstein
| title = Finite Groups
| id = {{MathSciNet | id = 81b:20002}}
| year = 1980
| publisher = Chelsea
| isbn = 0-8284-0301-5
| pages = 188–195
}}
 
[[Category:Finite groups]]

Revision as of 08:26, 5 March 2014


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