Extra special group
Encyclopedia
In group theory
, a branch of mathematics
, extra special groups are analogues of the Heisenberg group over finite field
s whose size is a prime. For each prime p and positive integer n there are exactly two (up to isomorphism) extra special groups of order
p1+2n. Extra special groups often occur in centralizers of involutions. The ordinary character theory
of extra special groups is well understood.
is called a p-group
if its order is a power of a prime p.
A p-group G is called extra special if its center
Z is cyclic of order p, and the quotient G/Z is a non-trivial elementary abelian
p-group.
Extra special groups of order p1+2n are often denoted by the symbol p1+2n. For example, 21+24 stands for an extra special group of order 225.
of extra special groups of order p3. This reduces the classification of extra special groups to that of extra special groups of order p3. The classification is often presented differently in the two cases p odd and p = 2, but a uniform presentation is also possible.
If n is a positive integer there are two extra special groups of order p1+2n, which for p odd are given by
The two extra special groups of order p1+2n are most easily distinguished by the fact that one has all elements of order at most p and the other has elements of order p2.
If n is a positive integer there are two extra special groups of order 21+2n, which are given by
The two extra special groups G of order 21+2n are most easily distinguished as follows. If Z is the center, then G/Z is a vector space over the field with 2 elements. It has a quadratic form q, where q is 1 if the lift of an element has order 4 in G, and 0 otherwise. Then the Arf invariant
of this quadratic form can be used to distinguish the two extra special groups. Equivalently, one can distinguish the groups by counting the number of elements of order 4.
M(p) and N(p) are non-isomorphic extra special groups of order p3 with center of order p generated by c. The two non-isomorphic extra special groups of order p1+2n are the central products of either n copies of M(p) or n−1 copies of M(p) and 1 copy of N(p). This is a special case of a classification of p-groups with cyclic centers and simple derived subgroups given in .
has structure 21+24.Co1, which means that it has a normal extra special subgroup of order 21+24, and the quotient is one of the Conway group
s.
, derived subgroup, and Frattini subgroup
are all equal are called special groups
. Infinite special groups whose derived subgroup has order p are also called extra special groups. The classification of countably infinite extra special groups is very similar to the finite case, , but for larger cardinalities even basic properties of the groups depend on delicate issues of set theory, some of which are exposed in . The nilpotent group
s whose center is cyclic and derived subgroup has order p and whose conjugacy classes are at most countably infinite are classified in . Finite groups whose derived subgroup has order p are classified in .
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 branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, extra special groups are analogues of the Heisenberg group over finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s whose size is a prime. For each prime p and positive integer n there are exactly two (up to isomorphism) extra special groups of order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
p1+2n. Extra special groups often occur in centralizers of involutions. The ordinary character theory
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
of extra special groups is well understood.
Definition
Recall that a finite groupFinite 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...
is called 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...
if its order is a power of a prime p.
A p-group G is called extra special if its center
Center (group theory)
In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
Z is cyclic of order p, and the quotient G/Z is a non-trivial 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....
p-group.
Extra special groups of order p1+2n are often denoted by the symbol p1+2n. For example, 21+24 stands for an extra special group of order 225.
Classification
Every extra special p-group has order p1+2n for some positive integer n, and conversely for each such number there are exactly two extra special groups up to isomorphism. A central product of two extra special p-groups is extra special, and every extra special group can be written as a central productCentral product
In mathematics, especially in the field of group theory, the central product is way of producing a group from two smaller groups. The central product is similar to the direct product, but in the central product two isomorphic central subgroups of the smaller groups are merged into a single central...
of extra special groups of order p3. This reduces the classification of extra special groups to that of extra special groups of order p3. The classification is often presented differently in the two cases p odd and p = 2, but a uniform presentation is also possible.
p odd
There are two extra special groups of order p3, which for p odd are given by- The group of triangular 3x3 matrices over the field with p elements, with 1's on the diagonal. This group has exponent p for p odd (but exponent 4 if p=2).
- The semidirect productSemidirect productIn mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
of a cyclic group of order p2 by a cyclic group of order p acting non-trivially on it. This group has exponent p2.
If n is a positive integer there are two extra special groups of order p1+2n, which for p odd are given by
- The central product of n extra special groups of order p3, all of exponent p. This extra special group also has exponent p.
- The central product of n extra special groups of order p3, at least one of exponent p2. This extra special group has exponent p2.
The two extra special groups of order p1+2n are most easily distinguished by the fact that one has all elements of order at most p and the other has elements of order p2.
p = 2
There are two extra special groups of order 8 = 23, which are given by- The dihedral groupDihedral groupIn mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
D8 of order 8, which can also be given by either of the two constructions in the section above for p = 2 (for p odd they given different groups, but for p = 2 they give the same group). This group has 2 elements of order 4. - The quaternion groupQuaternion groupIn group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication...
Q8 of order 8, which has 6 elements of order 4.
If n is a positive integer there are two extra special groups of order 21+2n, which are given by
- The central product of n extra special groups of order 8, an odd number of which are quaternion groups. The corresponding quadratic form (see below) has Arf invariant 1.
- The central product of n extra special groups of order 8, an even number of which are quaternion groups. The corresponding quadratic form (see below) has Arf invariant 0.
The two extra special groups G of order 21+2n are most easily distinguished as follows. If Z is the center, then G/Z is a vector space over the field with 2 elements. It has a quadratic form q, where q is 1 if the lift of an element has order 4 in G, and 0 otherwise. Then the Arf invariant
Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by when he started the systematic study of quadratic forms over arbitrary fields of characteristic2. The Arf invariant is the substitute, in...
of this quadratic form can be used to distinguish the two extra special groups. Equivalently, one can distinguish the groups by counting the number of elements of order 4.
All p
A uniform presentation of the extra special groups of order p1+2n can be given as follows. Define the two groups:
M(p) and N(p) are non-isomorphic extra special groups of order p3 with center of order p generated by c. The two non-isomorphic extra special groups of order p1+2n are the central products of either n copies of M(p) or n−1 copies of M(p) and 1 copy of N(p). This is a special case of a classification of p-groups with cyclic centers and simple derived subgroups given in .
Character theory
If G is an extra special group of order p1+2n, then its irreducible complex representations are given as follows:- There are exactly p2n irreducible representations of dimension 1. The center Z acts trivially, and the representations just correspond to the representations of the abelian group G/Z.
- There are exactly p−1 irreducible representations of dimension pn. There is one of these for each non-trivial character χ of the center, on which the center acts as multiplication by χ. The character values are given by pnχ on Z, and 0 for elements not in Z.
- If a nonabelian p-group G has less than p2-p nonlinear irreducible characters of minimal degree, it is extraspecial.
Examples
It is quite common for the centralizer of an involution in a finite simple group to contain a normal extra special subgroup. For example, the centralizer of an involution of type 2B in the monster groupMonster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
has structure 21+24.Co1, which means that it has a normal extra special subgroup of order 21+24, and the quotient is one of the Conway group
Conway group
In mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway.The largest of the Conway groups, Co1, of order...
s.
Generalizations
Groups whose centerCenter (group theory)
In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
, derived subgroup, and Frattini subgroup
Frattini subgroup
In mathematics, the Frattini subgroup Φ of a group G is the intersection of all maximal subgroups of G. For the case that G is the trivial group e, which has no maximal subgroups, it is defined by Φ = e...
are all equal are called special groups
Special group (finite group theory)
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...
. Infinite special groups whose derived subgroup has order p are also called extra special groups. The classification of countably infinite extra special groups is very similar to the finite case, , but for larger cardinalities even basic properties of the groups depend on delicate issues of set theory, some of which are exposed in . The nilpotent group
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...
s whose center is cyclic and derived subgroup has order p and whose conjugacy classes are at most countably infinite are classified in . Finite groups whose derived subgroup has order p are classified in .