View source for 3-transposition group

{{Short description|A group}}
In mathematical [[Group theory|group theory]], a '''3-transposition group''' is a [[Group (mathematics)|group]] generated by a conjugacy class of [[Involution (mathematics)|involutions]], called the '''3-transpositions''', such that the product of any two involutions from the conjugacy class has [[Order (group theory)|order]] at most 3. 

They were first studied by {{harvs|txt|first=Bernd |last=Fischer|year1=1964|year2=1970|year3= 1971}} who discovered the three [[Fischer group]]s as examples of 3-transposition groups.

==History==
{{harvtxt|Fischer|1964}} first studied 3-transposition groups in the special case when the product of any two distinct 3-transpositions has order 3. He showed that a finite group with this property is solvable, and has a (nilpotent) 3-group of index 2. {{harvtxt|Manin|1986}} used these groups to construct examples of non-abelian [[CH-quasigroup]]s and to describe the structure of commutative [[Moufang loop]]s of exponent 3.

==Fischer's theorem==
Suppose that ''G'' is a group  that is generated by a conjugacy class ''D'' of 3-transpositions and such that the [[P-core|2 and 3 cores]] ''O''<sub>2</sub>(''G'') and ''O''<sub>3</sub>(''G'') are both contained in the center ''Z''(''G'') of ''G''. Then {{harvtxt|Fischer|1971}} proved that up to isomorphism ''G''/''Z''(''G'') is one of the following groups and ''D'' is the image of the given conjugacy class:
*''G''/''Z''(''G'') is the trivial group.
*''G''/''Z''(''G'') is a [[Symmetric group|symmetric group]] S<sub>n</sub> for ''n''≥5, and ''D'' is the class of transpositions. (If ''n''=6  there is a second class of 3-transpositions).
*''G''/''Z''(''G'') is a [[Symplectic group|symplectic group]] Sp<sub>2''n''</sub>(2) with ''n''≥3 over the field of order 2, and ''D'' is the class of transvections. (When  ''n''=2 there is a second class of transpositions.)
*''G''/''Z''(''G'') is a projective special unitary group PSU<sub>''n''</sub>(2) with ''n''≥5, and ''D'' is the class of transvections
*''G''/''Z''(''G'') is an [[Orthogonal group|orthogonal group]] O<sup>μ</sup><sub>2''n''</sub>(2) with μ=±1 and ''n''≥4, and ''D'' is the class of transvections
*''G''/''Z''(''G'') is an index 2 subgroup PO<sub>''n''</sub><sup>μ,+</sup>(3) of the projective orthogonal group PO<sub>''n''</sub><sup>μ</sup>(3) (with μ=±1 and ''n''≥5) generated by the class ''D'' of reflections of norm +1 vectors.
*''G''/''Z''(''G'') is one of the three [[Fischer group]]s Fi<sub>22</sub>, Fi<sub>23</sub>, Fi<sub>24</sub>.
*''G''/''Z''(''G'') is one of two groups of the form Ω<sub>8</sub><sup>+</sup>(2).S<sub>3</sub> and PΩ<sub>8</sub><sup>+</sup>(3).S<sub>3</sub>, where Ω stands for the derived subgroup of the orthogonal group and S<sub>3</sub> is the group of diagram automorphisms for the D<sub>4</sub> Dynkin diagram.

The missing cases with ''n'' small above either do not satisfy the condition about 2 and 3 cores or have exceptional isomorphisms to other groups on the list.

==Important examples==

The group S<sub>''n''</sub> has order ''n''! and for ''n''>1 has a subgroup A<sub>''n''</sub> of index 2 that is simple if ''n''>4.

The symmetric group S<sub>''n''</sub> is a 3-transposition group for all ''n''>1. The 3-transpositions are the elements that exchange two points, and leaving each of the remaining points fixed. These elements are the [[Transposition (mathematics)|transpositions]] (in the usual sense) of S<sub>n</sub>. (For ''n''=6 there is a second class of 3-transpositions, namely the class of the elements of S<sub>6</sub> which are products of 3 disjoint transpositions.)

The symplectic group Sp<sub>2''n''</sub>(2) has order
:<math>2^{n^2}(2^2-1)(2^4-1)\cdots (2^{2n}-1)</math>
It is a 3-transposition group for all ''n''≥1. It is simple if ''n''>2, while for ''n''=1 it is S<sub>3</sub>, and for ''n''=2 it is S<sub>6</sub> with a simple subgroup of index 2, namely A<sub>6</sub>. The 3-transpositions are of the form ''x''↦''x''+(''x'',''v'')''v'' for non-zero ''v''.

The special unitary group SU<sub>''n''</sub>(2) has order
:<math>2^{n(n-1)/2}(2^2-1)(2^3+1)\cdots (2^n-(-1)^n)</math>
The projective special unitary group PSU<sub>''n''</sub>(2) is the quotient of the special unitary group SU<sub>''n''</sub>(2) by the subgroup ''M'' of all the scalar linear transformations in SU<sub>''n''</sub>(2). The subgroup ''M'' is the center of SU<sub>''n''</sub>(2). Also, ''M'' has order gcd(3,''n'').

The group PSU<sub>''n''</sub>(2) is simple if ''n''>3, while for ''n''=2 it is S<sub>3</sub> and for ''n''=3 it has the structure 3<sup>2</sup>:Q<sub>8</sub> (Q<sub>8</sub> = quaternion group).

Both SU<sub>''n''</sub>(2) and PSU<sub>''n''</sub>(2) are 3-transposition groups for ''n''=2 and for all ''n''≥4. The 3-transpositions of SU<sub>''n''</sub>(2) for ''n''=2 or ''n''≥4 are of the form ''x''↦''x''+(''x'',''v'')''v'' for non-zero vectors ''v'' of zero norm. The 3-transpositions of PSU<sub>''n''</sub>(2) for ''n''=2 or ''n''≥4 are the images of the 3-transpositions of SU<sub>''n''</sub>(2) under the natural quotient map from SU<sub>''n''</sub>(2) to PSU<sub>''n''</sub>(2)=SU<sub>''n''</sub>(2)/''M''.

The orthogonal group O<sub>2''n''</sub><sup>±</sup>(2) has order
:<math>2\times 2^{n(n-1)}(2^2-1)(2^4-1)\cdots (2^{2n-2}-1)(2^{n}\mp1)</math>
(Over fields of characteristic 2, orthogonal group in odd dimensions are isomorphic to symplectic groups.) It has an index 2 subgroup (sometimes denoted by Ω<sub>2''n''</sub><sup>±</sup>(2)), which is simple if ''n''>2.

The group O<sub>2''n''</sub><sup>μ</sup>(2) is a 3-transposition group for all ''n''>2 and μ=±1. The 3-transpositions are of the form ''x''↦''x''+(''x'',''v'')''v'' for vectors ''v'' such that ''Q''(v)=1, where ''Q'' is the underlying quadratic form for the orthogonal group.

The orthogonal groups O<sub>''n''</sub><sup>±</sup>(3) are the automorphism groups of quadratic forms ''Q'' over the field of 3 elements such that the discriminant of the bilinear form (''a'',''b'')=''Q''(''a''+''b'')−''Q''(''a'')−''Q''(''b'') is ±1. The group O<sub>''n''</sub><sup>μ,σ</sup>(3), where μ and σ are signs, is the subgroup of O<sub>''n''</sub><sup>μ</sup>(3) generated by reflections with respect to vectors ''v'' with ''Q''(''v'')=+1 if σ is +, and is the subgroup of O<sub>''n''</sub><sup>μ</sup>(3) generated by reflections with respect to vectors ''v'' with ''Q''(''v'')=-1 if σ is −.

For μ=±1 and σ=±1, let PO<sub>''n''</sub><sup>μ,σ</sup>(3)=O<sub>''n''</sub><sup>μ,σ</sup>(3)/''Z'', where ''Z'' is the group of all scalar linear transformations in O<sub>''n''</sub><sup>μ,σ</sup>(3). If ''n''>3, then ''Z'' is the center of O<sub>''n''</sub><sup>μ,σ</sup>(3).

For μ=±1, let Ω<sub>''n''</sub><sup>μ</sup>(3) be the derived subgroup of O<sub>''n''</sub><sup>μ</sup>(3). Let PΩ<sub>''n''</sub><sup>μ</sup>(3)= Ω<sub>''n''</sub><sup>μ</sup>(3)/''X'', where ''X'' is the group of all scalar linear transformations in Ω<sub>''n''</sub><sup>μ</sup>(3). If ''n''>2, then ''X'' is the center of Ω<sub>''n''</sub><sup>μ</sup>(3).

If ''n''=2''m''+1 is odd the two orthogonal groups O<sub>''n''</sub><sup>±</sup>(3) are isomorphic and have order
:<math>2\times 3^{m^2}(3^2-1)(3^4-1)\cdots(3^{2m}-1)</math>
and  O<sub>''n''</sub><sup>+,+</sup>(3) ≅ O<sub>''n''</sub><sup>−,−</sup>(3) (center order 1 for ''n''>3), and O<sub>''n''</sub><sup>−,+</sup>(3) ≅ O<sub>''n''</sub><sup>+,−</sup>(3) (center order 2 for ''n''>3), because the two quadratic forms are scalar multiples of each other, up to linear equivalence.

If ''n''=2''m'' is even the two orthogonal groups O<sub>''n''</sub><sup>±</sup>(3) have orders
:<math>2\times 3^{m(m-1)}(3^2-1)(3^4-1)\cdots(3^{2m-2}-1)(3^m\mp 1)</math>
and O<sub>''n''</sub><sup>+,+</sup>(3) ≅ O<sub>''n''</sub><sup>+,−</sup>(3), and O<sub>''n''</sub><sup>−,+</sup>(3) ≅ O<sub>''n''</sub><sup>−,−</sup>(3), because the two classes of transpositions are exchanged by an element of the general orthogonal group that multiplies the quadratic form by a scalar. If ''n''=2''m'', ''m''>1 and ''m'' is even, then the centre of O<sub>''n''</sub><sup>+,+</sup>(3) ≅ O<sub>''n''</sub><sup>+,−</sup>(3) has order 2, and the centre of O<sub>''n''</sub><sup>−,+</sup>(3) ≅ O<sub>''n''</sub><sup>−,−</sup>(3) has order 1. If ''n''=2''m'', ''m''>2 and ''m'' is odd, then the centre of O<sub>''n''</sub><sup>+,+</sup>(3) ≅ O<sub>''n''</sub><sup>+,−</sup>(3) has order 1, and the centre of O<sub>''n''</sub><sup>−,+</sup>(3) ≅ O<sub>''n''</sub><sup>−,−</sup>(3) has order 2.

If ''n''>3, and μ=±1 and σ=±1, the group O<sub>''n''</sub><sup>μ,σ</sup>(3) is a 3-transposition group. The 3-transpositions of the group O<sub>''n''</sub><sup>μ,σ</sup>(3) are of the form ''x''↦''x''−(''x'',''v'')''v''/Q(''v'')=''x''+(''x'', ''v'')/(''v'',''v'') for vectors ''v'' with ''Q''(''v'')=σ, where ''Q'' is the underlying quadratic form of O<sub>''n''</sub><sup>μ</sup>(3).

If ''n''>4, and μ=±1 and σ=±1, then O<sub>''n''</sub><sup>μ,σ</sup>(3) has index 2 in the orthogonal group O<sub>''n''</sub><sup>μ</sup>(3). The group O<sub>''n''</sub><sup>μ,σ</sup>(3) has a subgroup of index 2, namely Ω<sub>''n''</sub><sup>μ</sup>(3), which is simple modulo their centers (which have orders 1 or 2). In other words, PΩ<sub>''n''</sub><sup>μ</sup>(3) is simple.

If ''n''>4 is odd, and (μ,σ)=(+,+) or (−,−), then O<sub>''n''</sub><sup>μ,+</sup>(3) and PO<sub>''n''</sub><sup>μ,+</sup>(3) are both isomorphic to SO<sub>''n''</sub><sup>μ</sup>(3)=Ω<sub>''n''</sub><sup>μ</sup>(3):2, where SO<sub>''n''</sub><sup>μ</sup>(3) is the special orthogonal group of the underlying quadratic form ''Q''. Also, Ω<sub>''n''</sub><sup>μ</sup>(3) is isomorphic to PΩ<sub>''n''</sub><sup>μ</sup>(3), and is also non-abelian and simple.

If ''n''>4 is odd, and (μ,σ)=(+,−) or (−,+), then O<sub>''n''</sub><sup>μ,+</sup>(3) is isomorphic to Ω<sub>''n''</sub><sup>μ</sup>(3)×2, and O<sub>''n''</sub><sup>μ,+</sup>(3) is isomorphic to Ω<sub>''n''</sub><sup>μ</sup>(3). Also, Ω<sub>''n''</sub><sup>μ</sup>(3) is isomorphic to PΩ<sub>''n''</sub><sup>μ</sup>(3), and is also non-abelian and simple.

If ''n''>5 is even, and μ=±1 and σ=±1, then O<sub>''n''</sub><sup>μ,+</sup>(3) has the form Ω<sub>''n''</sub><sup>μ</sup>(3):2, and PO<sub>''n''</sub><sup>μ,+</sup>(3) has the form PΩ<sub>''n''</sub><sup>μ</sup>(3):2. Also, PΩ<sub>''n''</sub><sup>μ</sup>(3) is non-abelian and simple.

''Fi''<sub>22</sub> has order 2<sup>17</sup>.3<sup>9</sup>.5<sup>2</sup>.7.11.13 = 64561751654400 and is simple.

''Fi''<sub>23</sub> has order 2<sup>18</sup>.3<sup>13</sup>.5<sup>2</sup>.7.11.13.17.23 = 4089470473293004800 and is simple.

''Fi''<sub>24</sub> has order 2<sup>22</sup>.3<sup>16</sup>.5<sup>2</sup>.7<sup>3</sup>.11.13.17.23.29
and has a simple subgroup of index 2, namely ''Fi''<sub>24</sub>'.

==Isomorphisms and solvable cases==

There are numerous degenerate (solvable) cases and isomorphisms between 3-transposition groups of small degree as follows {{harv|Aschbacher|1997|loc=p.46}}:

===Solvable groups===

The following groups do not appear in the conclusion of Fisher's theorem as they are solvable (with order a power of 2 times a power of 3).

:<math>S_1=SU_1(2)=PSU_1(2)=O_1^{\pm,\pm}(3)=PO_1^{\pm,\pm}(3)=PO_1^{\pm,\mp}(3)</math> has order 1.
:<math>S_2=O_2^+(2)=O_1^{\pm,\mp}(3)=O_2^{+,\pm}(3)=PO_2^{+,\pm}(3)=PO_2^{-,\pm}(3)</math> has order 2, and it is a 3-transposition group.
:<math>O_2^{-,\pm}(3)=PO_3^{\pm,\mp}(3)=2^2</math> is elementary abelian of order 4, and it is '''not''' a 3-transposition group.
:<math>S_3=Sp_2(2)=SU_2(2)=PSU_2(2)=O_2^-(2)</math> has order 6, and it is a 3-transposition group.
:<math>O_3^{\pm,\mp}(3)=2^3</math> is elementary abelian of order 8, and it is '''not''' a 3-transposition group.
:<math>S_4=O_3^{\pm,\pm}(3)=PO_3^{\pm,\pm}(3)</math> has order 24, and it is a 3-transposition group.
:<math>PSU_3(2)=3^2:Q_8</math> has order 72, and it is '''not''' a 3-transposition group, where Q<sub>8</sub> denotes the quaternion group.
:<math>O_4^+(2)=(S_3\times S_3):2</math> has order 72, and it is '''not''' a 3-transposition group.
:<math>SU_3(2)=3^{1+2}:Q_8</math> has order 216, and it is '''not''' a 3-transposition group, where 3<sup>1+2</sup> denotes the extraspecial group of order 27 and exponent 3, and Q<sub>8</sub> denotes the quaternion group.
:<math> PO_4^{+,\pm}(3)=(A_4\times A_4):2</math> has order 288, and it is '''not''' a 3-transposition group.
:<math> O_4^{+,\pm}(3)=(SL_2(3)*SL_2(3)):2</math> has order 576, where * denotes the non-direct central product, and it is '''not''' a 3-transposition group.

===Isomorphisms===

There are several further isomorphisms involving groups in the conclusion of Fischer's theorem as follows. This list also identifies the Weyl groups of ADE Dynkin diagrams, which are all 3-transposition groups except W(D<sub>2</sub>)=2<sup>2</sup>, with groups on Fischer's list (W stands for Weyl group).

:<math>S_5=O_4^-(2)</math> has order 120, and the group is a 3-transposition group.
:<math>S_6=Sp_4(2)=O_4^{-,\pm}(3) = PO_4^{-,\pm}(3)</math> has order 720 (and 2 classes of 3-transpositions), and the group is a 3-transposition group.
:<math>S_8=O_6^+(2)=</math> has order 40320, and the group is a 3-transposition group.
:<math>W(E_6)=O_6^-(2)=O_5^{\pm,\pm}(3)=PO_5^{\pm,\pm}(3)</math> has order 51840, and the group is a 3-transposition group.
:<math>PO_5^{\pm,\mp}(3)=SU_4(2)=PSU_4(2)</math> has order 25920, and the group is a 3-transposition group.
:<math>W(E_7)=2\times Sp_6(2)</math> has order 2903040, and the group is a 3-transposition group.
:<math>W(E_8)=2.O_8^+(2) </math> has order 69672960, and the group is a 3-transposition group.
:<math>O_{4s}^{+,+}(3)=O_{4s}^{+,-}(3)=2.PO_{4s}^{+,+}(3)=2.PO_{4s}^{+,-}(3)</math> for all ''s''≥1, and the group is a 3-transposition group if ''s''≥2.
:<math>O_{4s}^{-,+}(3)=O_{4s}^{-,-}(3)=PO_{4s}^{-,+}(3)=PO_{4s}^{-,-}(3)</math> for all ''s''≥1, and the group is a 3-transposition group for all ''s''≥1.
:<math>O_{4s+2}^{+,+}(3)=O_{4s+2}^{+,-}(3)=2.PO_{4s+2}^{+,+}(3)=2.PO_{4s+2}^{+,-}(3)</math> for all ''s''≥0, and the group is a 3-transposition group for all ''s''≥0.
:<math>O_{4s+2}^{-,+}(3)=O_{4s+2}^{-,-}(3)=PO_{4s+2}^{-,+}(3)=PO_{4s+2}^{-,-}(3)</math> for all ''s''≥0, and the group is a 3-transposition group if ''s''≥1.
:<math>O_{2m+1}^{+,+}(3)=O_{2m+1}^{-,-}(3)=PO_{2m+1}^{+,+}(3)=PO_{2m+1}^{-,-}(3)</math> for all ''m''≥0, and the group is a 3-transposition group if ''m''≥1.
:<math>O_{2m+1}^{-,+}(3)=O_{2m+1}^{+,-}(3)=2\times PO_{2m+1}^{-,+}(3)=2\times PO_{2m+1}^{+,-}(3)</math> for all ''m''≥0, and the group is a 3-transposition group if ''m''=0 or ''m''≥2.
:<math>W(A_n)=S_{n+1}</math> for all ''n''≥1, and the group is a 3-transposition group for all n≥1.
:<math>W(D_n)=2^{n-1}.S_n</math> for all ''n''≥2, and the group is a 3-transposition group if n≥3.

==Proof==

The idea of the proof is as follows. Suppose that ''D'' is the class of 3-transpositions in ''G'', and ''d''∈''D'', and let ''H'' be the subgroup generated by the set ''D<sub>d</sub>'' of elements of ''D'' commuting with ''d''. Then ''D<sub>d</sub>'' is a set of 3-transpositions of ''H'', so the 3-transposition groups can be classified by induction on the order by finding all possibilities for ''G'' given any 3-transposition group ''H''. For simplicity assume that the derived group of ''G'' is perfect (this condition is satisfied by all but the two groups involving triality automorphisms.)

*If ''O''<sub>3</sub>(''H'') is not contained in ''Z''(''H'') then ''G'' is the symmetric group ''S''<sub>5</sub>
*If ''O''<sub>2</sub>(''H'') is not contained in ''Z''(''H'') then ''L''=''H''/''O''<sub>2</sub>(''H'') is a 3-transposition group, and ''L''/''Z''(''L'') is either of type Sp(2''n'', 2) in which case ''G''/''Z''(''G'') is of type Sp<sub>2''n''+2</sub>(2), or  of type PSU<sub>''n''</sub>(2) in which case ''G''/''Z''(''G'') is of type PSU<sub>''n''+2</sub>(2)
*If ''H''/''Z''(''H'') is of type ''S<sub>n</sub>'' then either ''G'' is of type ''S''<sub>''n''+2</sub> or ''n'' = 6 and ''G'' is of type O<sub>6</sub><sup>−</sup>(2)
*If ''H''/''Z''(''H'') is of type Sp<sub>2''n''</sub>(2) with 2''n'' ≥ 6 then ''G'' is of type O<sub>2''n''+2</sub><sup>μ</sup>(2)
*''H''/''Z''(''H'') cannot be of type O<sub>2''n''</sub><sup>μ</sup>(2) for ''n'' ≥ 4.
*If ''H''/''Z''(''H'') is of type PO<sub>''n''</sub><sup>μ, π</sup>(3) for ''n''>4 then ''G'' is of type PO<sub>''n''+1</sub><sup>−μπ, π</sup>(3).
*If ''H''/''Z''(''H'') is of type PSU<sub>''n''</sub>(2) for ''n'' ≥ 5 then ''n'' = 6 and ''G'' is of type Fi<sub>22</sub> (and ''H'' is an exceptional double cover of PSU<sub>6</sub>(2))
*If ''H''/''Z''(''H'') is of type Fi<sub>22</sub> then ''G'' is of type Fi<sub>23</sub> and ''H'' is a double cover of Fi<sub>22</sub>.
*If ''H''/''Z''(''H'') is of type Fi<sub>23</sub> then ''G'' is of type Fi<sub>24</sub> and ''H'' is the product of Fi<sub>23</sub> and a group of order 2.
*''H''/''Z''(''H'') cannot be of type Fi<sub>24</sub>.

==3-transpositions and graph theory==
It is fruitful to treat 3-transpositions as vertices of a [[Graph theory|graph]]. Join the pairs that do not commute, i. e. have a product of order 3. The graph is connected unless the group has a direct product decomposition. The graphs corresponding to the smallest symmetric groups are familiar graphs. The 3 transpositions of ''S''<sub>3</sub> form a triangle. The 6 transpositions of ''S''<sub>4</sub> form an octahedron. The 10 transpositions of ''S''<sub>5</sub> form the complement of the [[Petersen graph]].

The symmetric group ''S<sub>n</sub>'' can be generated by ''n''–1 transpositions: (1 2), (2 3), ..., (''n''−1 ''n'') and the graph of this generating set is a straight line. It embodies sufficient relations to define the group ''S<sub>n</sub>''.<ref>{{citation |first=L. E. |last=Dickson |title=Linear Groups: With an Exposition of the Galois Field Theory |url=https://books.google.com/books?id=7TKuQu80UAsC&pg=PA287 |year=2003 |isbn=978-0-486-49548-4 |page=287 |origyear=1900}}</ref>

==References==
{{reflist}}
{{refbegin}}
*{{Citation | last1=Aschbacher | first1=Michael |  title=3-transposition groups | publisher={{wipe|Cambridge University Press}} | series=Cambridge Tracts in Mathematics | isbn=978-0-521-57196-8 | mr=1423599 | year=1997 | volume=124 | url=http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511759413 | access-date=2010-12-06 | archive-date=2016-03-04 | archive-url=https://web.archive.org/web/20160304045542/http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511759413 | url-status=dead }} contains a complete proof of Fischer's theorem.
*{{Citation | last1=Fischer | first1=Bernd | title=Distributive Quasigruppen endlicher Ordnung | doi=10.1007/BF01111162 | mr=0160845 | year=1964 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=83 | issue=4 | pages=267–303| s2cid=123008891 }}
*{{Citation | last1=Fischer | first1=Bernd | title=Finite groups generated by 3-transpositions  | year=1970 | publisher=Mathematics Institute, University of Warwick|place =Coventry|series= preprint}} The first part of this preprint (4 of 19 sections) was published as {{Citation | last1=Fischer | first1=Bernd | title=Finite groups generated by 3-transpositions. I | mr=0294487 | year=1971 | journal=[[Inventiones Mathematicae]] | volume=13 |issue=3 | pages=232–246 |doi=10.1007/BF01404633 |bibcode=1971InMat..13..232F | s2cid=120817150 }} The later part with the construction of the Fischer groups is still unpublished (as of 2014).
*{{Citation | last1=Manin | first1=Yuri Ivanovich |  title=Cubic forms | origyear=1972 | url=https://books.google.com/books?id=W03vAAAAMAAJ | publisher=North-Holland | location=Amsterdam | edition=2nd | series=North-Holland Mathematical Library | isbn=978-0-444-87823-6 | mr=833513 | year=1986 | volume=4 }}
*{{citation|mr=0733341
|last=Weiss |first= Richard
|title=On Fischer's characterization of Sp<sub>2n</sub>(2) and U<sub>n</sub>(2)
|journal=Communications in Algebra |volume=11 |year=1983 |issue= 22 |pages= 2527–54 |doi=10.1080/00927878308822979 }}
*{{citation |mr=0778676 |last=Weiss |first=Richard |title=A uniqueness lemma for groups generated by 3-transpositions |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=97 |year=1985 |issue=3 |pages=421–431 |doi=10.1017/S030500410006299X |url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2089284&fileId=S030500410006299X |bibcode=1985MPCPS..97..421W |s2cid=123397959 }}
{{refend}}

[[Category:Finite groups]]

{{Sourceattribution|3-transposition group}}