Special group (finite group theory)
Encyclopedia
In group theory
, a discipline within abstract algebra
, a special group is a finite group
of prime power
order that is either elementary abelian
itself or of class 2 with its derived group, its center, and its Fitting subgroup
all equal and elementary abelian . A special group of order pn that has class 2 and whose derived group has order p is called an extra special group
.
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 discipline within abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a special group is a finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
of prime power
Prime power
In mathematics, a prime power is a positive integer power of a prime number.For example: 5=51, 9=32 and 16=24 are prime powers, while6=2×3, 15=3×5 and 36=62=22×32 are not...
order that is either elementary abelian
Elementary Abelian group
In group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order p, where p is a prime; in particular it is a p-group....
itself or of class 2 with its derived group, its center, and its Fitting subgroup
Fitting subgroup
In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable...
all equal and elementary abelian . A special group of order pn that has class 2 and whose derived group has order p is called an extra special group
Extra special group
In group theory, a branch of mathematics, extra special groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime p and positive integer n there are exactly two extra special groups of order p1+2n. Extra special groups often occur in centralizers of...
.