Elementary group
Encyclopedia
In algebra, more specifically group theory
, a p-elementary group is a direct product
of a finite cyclic group
of order relatively prime to p and a p-group
. A finite group is an elementary group if it is p-elementary for some prime number p. An elementary group is nilpotent
.
Brauer's theorem on induced characters
states that a character on a finite group is a linear combination with integer coefficients of characters induced from elementary subgroups.
More generally, a finite group is called a p-hyperelementary if it has the extension
where
is cyclic of order prime to p and P is a p-group. Not every hyperelementary group is elementary: for instance the non-abelian group of order 6 is 2-hyperelementary, but not 2-elementary. The term "hyperelementary" was introduced by G. Segal.
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a p-elementary group is a direct product
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
of a finite cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
of order relatively prime to p and a p-group
P-group
In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...
. A finite group is an elementary group if it is p-elementary for some prime number p. An elementary group is nilpotent
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
.
Brauer's theorem on induced characters
Brauer's theorem on induced characters
Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, which is, in turn, part of the representation theory of a finite group...
states that a character on a finite group is a linear combination with integer coefficients of characters induced from elementary subgroups.
More generally, a finite group is called a p-hyperelementary if it has the extension
where
is cyclic of order prime to p and P is a p-group. Not every hyperelementary group is elementary: for instance the non-abelian group of order 6 is 2-hyperelementary, but not 2-elementary. The term "hyperelementary" was introduced by G. Segal.Reference
- Arthur Bartels, Wolfgang Lück, Induction Theorems and Isomorphism Conjectures for K- and L-Theory
- G. Segal, The representation-ring of a compact Lie group
- J.P. Serre, "Linear representations of finite groups". Graduate Texts in Mathematics, vol. 42, Springer-Verlag, New York, Heidelberg, Berlin, 1977,