In
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, more specifically
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, a ''p''-elementary group is a
direct product
In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...
of a
finite
Finite may refer to:
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect
* "Finite", a song by Sara Gr ...
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
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
...
relatively prime to ''p'' and a
''p''-group. A finite group is an elementary group if it is ''p''-elementary for some
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
''p''. An elementary group is
nilpotent
In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term, along with its sister Idempotent (ring theory), idem ...
.
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 ''G'' 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.
See also
*
Elementary abelian group
In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same order. This common order must be a prime number, and the elementary abelian groups in whic ...
References
* 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,
Properties of groups
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