Extra special group

In group theory, a branch of mathematics, extra special groups are analogues of the Heisenberg group over fields of prime order "p".

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 extraspecial 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 "p"^1+2"n" are often denoted by the symbol "p"^1+2"n". For example, 2¹⁺²⁴ stands for an extraspecial group of order 2²⁵.

Classification

Every extra special "p"-group has order "p"^1+2"n" for some positive integer "n", and conversely for each such number there are exactly two extraspecial groups up to isomorphism. A central product of two extraspecial "p"-groups is extraspecial, and every extraspecial group can be written as a central product of extra special groups of order "p"³. SO this reduces the classification of extraspecial groups to that of extraspecial groups of order "p"³. The classification is different in the two cases "p" odd and "p" = 2.

"p" odd

There are two extraspecial groups of order "p"³, which for "p" odd are given by
* The group of triangular 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 "p"² by a cyclic group of order "p" acting non-trivially on it. This group has exponent "p"².

If "n" is a positive integer there are two extraspecial groups of order "p"^1+2"n", which for "p" odd are given by
*The central product of "n" extraspecial groups of order "p"³, all of exponent "p". This extraspecial group also has exponent "p".
*The central product of "n" extraspecial groups of order "p"³, at least one of exponent "p"². This extraspecial group has exponent "p"².

The two extraspecial groups of order "p"^1+2"n" are most easily distinguished by the fact that one has all elements of order at most "p" and the other has elements of order "p"².

="p" = 2=

There are two extraspecial groups of order 8 = "2"³, which are given by
* The dihedral group "D"₈ 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 "Q"₈ of order 8, which has 6 elements of order 4.

If "n" is a positive integer there are two extraspecial groups of order "2"^1+2"n", which are given by
*The central product of "n" extraspecial 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" extraspecial groups of order 8, an even number of which are quaternion groups. The corresponding quadratic form (see below) has Arf invariant 0.

The two extraspecial groups "G" of order "2"^1+2"n" 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 extraspecial groups. (Equivalently, one can distinguish the groups by counting the number of elements of order 4.)

Character theory

If "G" is an extraspecial group of order "p"^1+2"n", then its irreducible complex representations are given as follows:
*There are exactly "p"^2"n" 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 "p"^"n". 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 "p"^"n"χ on "Z", and 0 for elements not in "Z".

Examples

It is quite common for the centralizer of an involution in a finite simple group to contain a normal extraspecial subgroup. For example, the centralizer of an involution of type 2B in the monster group has structure 2¹⁺²⁴.Co_{1, which means that it has a normal extraspecial subgroup of order 2¹⁺²⁴, and the quotient is one of the Conway groups.}

References

*D. Gorenstein, "Finite groups" ISBN 0-8284-0301-5