# Talk:Extra special group

## Detection

It would be convenient to have a quick method of detection whether an extra special group was of positive or negative type. The article currently suggests a method that is not a very good idea for p=2.

For odd primes, one can consider the orders of the elements of a generating set (all order p iff positive type), but for p=2 things are more complicated. A group of negative type cannot be generated by elements of order 2, while a group of positive type always can be. However, a given set of generators of a group of positive type may include elements of order 4.

The suggestion in the article is to just count the elements of order 4 (for reference, the positive type of order 2^(2n+1) has 4^n − 2^n elements of order 4, and the negative type has 4^n + 2^n elements of order 4). This requires time at least exponential in n, so is not very appealing (the method for odd p would typically require time linear in n). Again, comparing the case of odd p, the positive type has 0 elements of order p^2 while the negative type has (p−1)p^n elements of order p^2.

Can the quadratic form suggested in the article be expressed in terms of the bilinear form giving the symplectic structure on G/Z? If so, it might be faster to use linear algebra, since the bilinear form can be constructed typically in time quadratic in n, and surely any linear algebra needed to compute the Arf invariant is polynomial in n. JackSchmidt (talk) 22:58, 24 May 2009 (UTC)

## Weyl-Heisenberg group is missing?

I do not understand this statement from section p=2:

There are two extra special groups of order 8

Here the article considers the dihedral and the quaternion group. I always thought that the Heisenberg groups for p=2 are also extra special p-groups. I know this group is, in general, not isomorphic to the former ones, but I do not know if this holds for the particular case of order 8, here considered. If there is an isomorphism it should be mentioned and documented since Heisenberg groups are important.

Also, this family of groups should be cited somewhere.

Last thing, I know that there are more classes of 2-groups than dihedral groups and quaternions, but I do not know much about them. They are metioned in Fast Quantum Fourier Transforms for a Class of Non-abelian Groups. Their original reference is, unfortunately, only in German.

Garrapito (talk) 14:28, 2 March 2011 (UTC)

As mention on Heisenberg group, the Heisenberg group over the field Z/2Z is the dihedral group of order 8. Heisenberg groups of larger dimension over the field Z/2Z are central products of dihedral groups of order 8, and so extra-special groups. Dihedral groups (or quaternion groups) of order larger than 8 are not extra-special. Heisenberg groups over rings other than Z/pZ are not extra-special, though they can have analogous properties. For any prime p and positive integer n, there are exactly two extra-special groups of order p^(2n+1). For p=2 and n=1, those two groups are the dihedral group of order 8 (aka the Heisenberg 2-group) and the quaternion group of order 8. JackSchmidt (talk) 18:36, 11 March 2011 (UTC)
Thank you! Garrapito (talk) 23:15, 21 March 2011 (UTC)