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| In [[group theory]], a branch of [[mathematics]], '''extra special groups''' are analogues of the [[Heisenberg group]] over [[finite field]]s 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 (group theory)|order]] ''p''<sup>1+2''n''</sup>. Extra special groups often occur in centralizers of involutions. The ordinary [[character theory]] of extra special groups is well understood.
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| ==Definition==
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| Recall that a [[finite group]] is called a [[p-group|''p''-group]] if its order is a power of a prime ''p''.
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| A ''p''-group ''G'' is called '''extra special''' if its [[center (group theory)|center]] ''Z'' is cyclic of order ''p'', and the quotient ''G''/''Z'' is a non-trivial [[elementary abelian group|elementary abelian]] ''p''-group.
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| Extra special groups of order ''p''<sup>1+2''n''</sup> are often denoted by the symbol ''p''<sup>1+2''n''</sup>. For example, 2<sup>1+24</sup> stands for an extra special group of order 2<sup>25</sup>.
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| ==Classification==
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| Every extra special ''p''-group has order ''p''<sup>1+2''n''</sup> 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 ''p''<sup>3</sup>. This reduces the classification of extra special groups to that of extra special groups of order ''p''<sup>3</sup>. The classification is often presented differently in the two cases ''p'' odd and ''p'' = 2, but a uniform presentation is also possible.
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| ===''p'' odd===
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| There are two extra special groups of order ''p''<sup>3</sup>, which for ''p'' odd are given by
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| * 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).
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| * The [[semidirect product]] of a cyclic group of order ''p''<sup>2</sup> by a cyclic group of order ''p'' acting non-trivially on it. This group has exponent ''p''<sup>2</sup>.
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| If ''n'' is a positive integer there are two extra special groups of order ''p''<sup>1+2''n''</sup>, which for ''p'' odd are given by
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| *The central product of ''n'' extra special groups of order ''p''<sup>3</sup>, all of exponent ''p''. This extra special group also has exponent ''p''.
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| *The central product of ''n'' extra special groups of order ''p''<sup>3</sup>, at least one of exponent ''p''<sup>2</sup>. This extra special group has exponent ''p''<sup>2</sup>.
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| The two extra special groups of order ''p''<sup>1+2''n''</sup> 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''<sup>2</sup>.
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| ===''p'' = 2===
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| There are two extra special groups of order 8 = ''2''<sup>3</sup>, which are given by
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| * The [[dihedral group]] ''D''<sub>8</sub> 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.
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| *The [[quaternion group]] ''Q''<sub>8</sub> of order 8, which has 6 elements of order 4.
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| If ''n'' is a positive integer there are two extra special groups of order ''2''<sup>1+2''n''</sup>, which are given by
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| *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.
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| *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.
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| The two extra special groups ''G'' of order ''2''<sup>1+2''n''</sup> 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.
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| === All ''p'' ===
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| A uniform presentation of the extra special groups of order ''p''<sup>1+2''n''</sup> can be given as follows. Define the two groups:
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| * <math> M(p) = \langle a,b,c : a^p = b^p = 1, c^p = 1, ba=abc, ca=ac, cb=bc \rangle </math>
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| * <math> N(p) = \langle a,b,c : a^p = b^p = c, c^p = 1, ba=abc, ca=ac, cb=bc \rangle </math>
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| ''M''(''p'') and ''N''(''p'') are non-isomorphic extra special groups of order ''p''<sup>3</sup> with center of order ''p'' generated by ''c''. The two non-isomorphic extra special groups of order ''p''<sup>1+2''n''</sup> 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 {{harv|Newman|1960}}.
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| ==Character theory==
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| If ''G'' is an extra special group of order ''p''<sup>1+2''n''</sup>, then its irreducible complex representations are given as follows:
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| *There are exactly ''p''<sup>2''n''</sup> irreducible representations of dimension 1. The center ''Z'' acts trivially, and the representations just correspond to the representations of the abelian group ''G''/''Z''.
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| *There are exactly ''p''−1 irreducible representations of dimension ''p''<sup>''n''</sup>. 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''<sup>''n''</sup>χ on ''Z'', and 0 for elements not in ''Z''.
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| * If a nonabelian ''p''-group ''G'' has less than ''p<sup>2</sup>-p'' nonlinear irreducible characters of minimal degree, it is extraspecial.
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| ==Examples==
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| 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 2<sup>1+24</sup>.Co<sub>1</sup>, which means that it has a normal extra special subgroup of order 2<sup>1+24</sup>, and the quotient is one of the [[Conway group]]s.
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| == Generalizations ==
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| Groups whose [[center (group theory)|center]], [[derived subgroup]], and [[Frattini subgroup]] are all equal are called '''[[special group (finite group theory)|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, {{harv|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 {{harv|Shelah|Steprãns|1987}}. The [[nilpotent group]]s whose center is cyclic and derived subgroup has order ''p'' and whose conjugacy classes are at most countably infinite are classified in {{harv|Newman|1960}}. Finite groups whose derived subgroup has order ''p'' are classified in {{harv|Blackburn|1999}}.
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| ==References==
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| * {{Citation | last1=Blackburn | first1=Simon R. | title=Groups of prime power order with derived subgroup of prime order | doi=10.1006/jabr.1998.7909 | mr=1706841 | year=1999 | journal=Journal of Algebra | issn=0021-8693 | volume=219 | issue=2 | pages=625–657}}
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| * {{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=Finite Groups | publisher=Chelsea | location=New York | isbn=978-0-8284-0301-6 | mr=81b:20002 | year=1980 }}
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| * {{Citation | last1=Newman | first1=M. F. | title=On a class of nilpotent groups | doi=10.1112/plms/s3-10.1.365 | mr=0120278 | year=1960 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=10 | pages=365–375}}
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| * {{Citation | last1=Shelah | first1=Saharon | author1-link=Saharon Shelah | last2=Steprãns | first2=Juris | title=Extraspecial p-groups | doi=10.1016/0168-0072(87)90041-8 | mr=887554 | year=1987 | journal=Annals of Pure and Applied Logic | issn=0168-0072 | volume=34 | issue=1 | pages=87–97}}
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| [[Category:P-groups]]
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