Quasi-dihedral Group
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the quasi-dihedral groups, also called semi-dihedral groups, are certain
non-abelian group In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗  ...
s of
order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...
a power of 2. For every positive
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''n'' greater than or equal to 4, there are exactly four
isomorphism class In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them ...
es of non-abelian
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
s of order 2''n'' which have a cyclic
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
of
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2. Two are well known, the
generalized quaternion group In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication. It is given by the group presentation :\mathrm_8 ...
and the
dihedral group In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
. One of the remaining two groups is often considered particularly important, since it is an example of a 2-group of maximal nilpotency class. In Bertram Huppert's text ''Endliche Gruppen'', this group is called a "Quasidiedergruppe". In
Daniel Gorenstein Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician best remembered for his contribution to the classification of finite simple groups. Gorenstein mastered calculus at age 12 and subsequently matriculated at ...
's 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
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for this group: :\langle r,s \mid r^ = s^2 = 1,\ srs = r^\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). When this group has order 16, Dummit and Foote refer to this group as the "modular group of order 16", as its lattice of subgroups is modular. In this article this group will be called the modular maximal-cyclic group of order 2^n. Its presentation is: :\langle r,s \mid r^ = s^2 = 1,\ srs = r^\rangle\,\!. Both these two groups and the dihedral group are
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ...
s of a cyclic group <''r''> of order 2''n''−1 with a cyclic group <''s''> of order 2. Such a non-abelian semidirect product is uniquely determined by an element of order 2 in the
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of the
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\mathbb/2^\mathbb and there are precisely three such elements, 2^-1, 2^-1, and 2^+1, corresponding to the dihedral group, the quasidihedral, and the modular maximal-cyclic group. The generalized quaternion group, the dihedral group, and the quasidihedral group of order 2''n'' all have nilpotency class ''n'' − 1, and are the only isomorphism classes of groups of order 2''n'' with nilpotency class ''n'' − 1. The groups of order ''p''''n'' and nilpotency class ''n'' − 1 were the beginning of the classification of all ''p''-groups via coclass. The modular maximal-cyclic group of order 2''n'' always has nilpotency class 2. This makes the modular maximal-cyclic group less interesting, since most groups of order ''p''''n'' 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 In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal s ...
has index 4. The
Alperin–Brauer–Gorenstein theorem In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathedA 2-group is wreathed if it is a nonabelian semidirect product of a maximal subgroup that is a direct product of two cyc ...
classifies the
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
s, and to a degree the
finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
s, with quasidihedral Sylow 2-subgroups.


Examples

The Sylow 2-subgroups of the following groups are quasidihedral: *PSL3(F''q'') for ''q'' ≡ 3 mod 4, *PSU3(F''q'') for ''q'' ≡ 1 mod 4, *the
Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objec ...
M11, *GL2(F''q'') for ''q'' ≡ 3 mod 4.


References

* * * {{cite book , last = Gorenstein , first = D. , authorlink = Daniel Gorenstein , title = Finite Groups , mr=569209 , year = 1980 , publisher = Chelsea , isbn = 0-8284-0301-5 , pages = 188–195 Finite groups