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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 ''p''1+2''n''. 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 ''p''1+2''n'' are often denoted by the symbol ''p''1+2''n''. For example, 21+24 stands for an extra special group of order 225.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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