Extraspecial 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.

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.

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.

There are two extra special groups of order p³, which for p odd are given by

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.

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