Extra special group

In group theory, a branch of abstract algebra, 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 (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.

Definition

Recall that a finite group 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.

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 product 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 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 group 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 group 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 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:

$M(p)=\langle a,b,c:a^{p}=b^{p}=1,c^{p}=1,ba=abc,ca=ac,cb=bc\rangle$
$N(p)=\langle a,b,c:a^{p}=b^{p}=c,c^{p}=1,ba=abc,ca=ac,cb=bc\rangle$

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 (Newman 1960).

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 group 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 groups.

Generalizations

Groups whose center, 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, (Newman 1960), but for larger cardinalities even basic properties of the groups depend on delicate issues of set theory, some of which are exposed in (Shelah & Steprãns 1987). The nilpotent groups whose center is cyclic and derived subgroup has order p and whose conjugacy classes are at most countably infinite are classified in (Newman 1960). Finite groups whose derived subgroup has order p are classified in (Blackburn 1999).

References

Blackburn, Simon R. (1999), "Groups of prime power order with derived subgroup of prime order", Journal of Algebra, 219 (2): 625–657, ISSN 0021-8693, MR 1706841, doi:10.1006/jabr.1998.7909
Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 81b:20002
Newman, M. F. (1960), "On a class of nilpotent groups", Proceedings of the London Mathematical Society. Third Series, 10: 365–375, ISSN 0024-6115, MR 0120278, doi:10.1112/plms/s3-10.1.365
Shelah, Saharon; Steprāns, Juris (1987), "Extraspecial p-groups", Annals of Pure and Applied Logic, 34 (1): 87–97, ISSN 0168-0072, MR 887554, doi:10.1016/0168-0072(87)90041-8

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