In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain 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 non-abelian groups of order 2n which have a cyclic subgroup of index 2. Two are well known, the generalized quaternion group and the dihedral group. 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'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 presentation for this group:
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 Mm(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 . Its presentation is:
Both these two groups and the dihedral group are semidirect products of a cyclic group of order 2n−1 with a cyclic group of order 2. Such a non-abelian semidirect product is uniquely determined by an element of order 2 in the group of units of the ring and there are precisely three such elements, , , and , 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 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 modular maximal-cyclic group of order 2n always has nilpotency class 2.
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Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field K of positive characteristic p, necessarily a prime number. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.
In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p. Abelian p-groups are also called p-primary or simply primary. A finite group is a p-group if and only if its order (the number of its elements) is a power of p.
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 where e is the identity element and commutes with the other elements of the group. Another presentation of Q8 is The quaternion group Q8 has the same order as the dihedral group D4, but a different structure, as shown by their Cayley and cycle graphs: In the diagrams for D4, the group elements are marked with their action on a letter F in the defining representation R2.
This is an introduction to modern algebra: groups, rings and fields.
This course presents geometric constructions of irreducible representations of semi-simple Lie Algebras and their Weyl groups by means of Springer theory.
We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of Erdmann's algebras of dihedral, semi-dihedral and quaternion type and obtain as byproduct the validity of ...
We provide characterizations of p-nilpotency for fusion systems and p-local finite groups that are inspired by known result for finite groups. In particular, we generalize criteria by Atiyah, Brunetti, Frobenius, Quillen, Stammbach and Tate. ...
Springer2014
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Let G be a finite group with a Sylow 2-subgroup P which is either quaternion or semi-dihedral. Let k be an algebraically closed field of characteristic 2. We prove the existence of exotic endotrivial kG-modules, whose restrictions to P are isomorphic to th ...