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A [[Simple finite group|simple finite group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s0915101.png" />, a member of the infinite series of simple groups, discovered by M. Suzuki.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s0915102.png" /> be a natural number, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s0915103.png" /> be the finite field with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s0915104.png" /> elements, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s0915105.png" /> be an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s0915106.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s0915107.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s0915108.png" />. The Suzuki group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s0915109.png" /> is then generated by the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151010.png" /> consisting of all diagonal matrices of order 4 with diagonal elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151013.png" />), the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151014.png" /> consisting of all triangular matrices of the form
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151015.png" /></td> </tr></table>
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A [[Simple finite group|simple finite group]]  $  \mathop{\rm Sz} ( q) $,
 +
a member of the infinite series of simple groups, discovered by M. Suzuki.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151016.png" />, and the matrix
+
Let  $  n $
 +
be a natural number, let  $  F $
 +
be the finite field with  $  q = 2  ^ {2n+} 1 $
 +
elements, and let  $  \theta $
 +
be an automorphism of  $  F $
 +
such that  $  \alpha ^ {\theta  ^ {2} } = \alpha  ^ {2} $
 +
for any  $  \alpha \in F $.
 +
The Suzuki group  $  \mathop{\rm Sz} ( q) $
 +
is then generated by the subgroup  $  T $
 +
consisting of all diagonal matrices of order 4 with diagonal elements  $  \lambda ^ {1+ \theta } , \lambda , \lambda  ^ {-} 1 , ( \lambda ^ {1+ \theta } )  ^ {-} 1 $(
 +
$  \lambda \in F $,
 +
$  \lambda \neq 0 $),  
 +
the subgroup  $  U $
 +
consisting of all triangular matrices of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151017.png" /></td> </tr></table>
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$$
 +
\left \|
 +
\begin{array}{cccc}
 +
1  & 0 & 0  & 0  \\
 +
\alpha  & 1  & 0  & 0  \\
 +
\alpha ^ {1+ \theta } + \beta  &\alpha  ^  \theta  & 1  & 0  \\
 +
\alpha ^ {2+ \theta } + \alpha \beta + \beta  ^  \theta  &\beta  &\alpha  & 1  \\
 +
\end{array}
 +
\right \|
 +
$$
  
The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151018.png" /> is a Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151019.png" />-subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151020.png" />; it is a [[Suzuki-2-group|Suzuki <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151021.png" />-group]]. The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151022.png" /> coincides with the normalizer of the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151023.png" />. The permutation representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151024.png" /> on the cosets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151025.png" /> is doubly transitive; its degree is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151026.png" />. The order of the Suzuki group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151027.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151028.png" />, and is not divisible by 3. On the other hand, any non-Abelian finite simple group whose order is not divisible by 3 is isomorphic to a Suzuki group. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151029.png" /> is a maximal subgroup of the symplectic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151030.png" /> and is the centralizer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151031.png" /> of an automorphism of order 2 of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151032.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151033.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151034.png" /> — the twisted analogue of a [[Chevalley group|Chevalley group]] of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151035.png" /> over the field with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151036.png" /> elements.
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$  ( \alpha , \beta \in F  ) $,
 +
and the matrix
 +
 
 +
$$
 +
\left \|
 +
\begin{array}{cccc}
 +
0  & 0  & 0  & 1  \\
 +
0  & 0  & 1  & 0  \\
 +
0  & 1  & 0  & 0  \\
 +
1  & 0  & 0  & 0  \\
 +
\end{array}
 +
\right \| .
 +
$$
 +
 
 +
The subgroup $  U $
 +
is a Sylow $  2 $-
 +
subgroup of the group $  \mathop{\rm Sz} ( q) $;  
 +
it is a [[Suzuki-2-group|Suzuki $  2 $-
 +
group]]. The subgroup $  UT $
 +
coincides with the normalizer of the subgroup $  U $.  
 +
The permutation representation of the group $  \mathop{\rm Sz} ( q) $
 +
on the cosets of $  UT $
 +
is doubly transitive; its degree is equal to $  q  ^ {2} + 1 $.  
 +
The order of the Suzuki group $  \mathop{\rm Sz} ( q) $
 +
is equal to $  q  ^ {2} ( q- 1)( q  ^ {2} + 1) $,  
 +
and is not divisible by 3. On the other hand, any non-Abelian finite simple group whose order is not divisible by 3 is isomorphic to a Suzuki group. The group $  \mathop{\rm Sz} ( q) $
 +
is a maximal subgroup of the symplectic group $  \mathop{\rm Sp} ( 4, q) $
 +
and is the centralizer in $  \mathop{\rm Sp} ( 4, q) $
 +
of an automorphism of order 2 of the group $  \mathop{\rm Sp} ( 4, q) = B _ {2} ( q) $.  
 +
In other words, $  \mathop{\rm Sz} ( q) $
 +
is isomorphic to $  {}  ^ {2} B _ {2} ( q) $—  
 +
the twisted analogue of a [[Chevalley group|Chevalley group]] of type $  B _ {2} $
 +
over the field with $  q $
 +
elements.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Suzuki,  "On a class of doubly transitive groups"  ''Ann. of Math.'' , '''75''' :  1  (1962)  pp. 105–145</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.W. Carter,  "Simple groups of Lie type" , Wiley (Interscience)  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Suzuki,  "On a class of doubly transitive groups"  ''Ann. of Math.'' , '''75''' :  1  (1962)  pp. 105–145</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.W. Carter,  "Simple groups of Lie type" , Wiley (Interscience)  (1972)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
There is in fact precisely one automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151037.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151038.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151039.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151040.png" />. It is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151041.png" />.
+
There is in fact precisely one automorphism $  \theta $
 +
of $  F $
 +
such that $  \theta  ^ {2} ( \alpha ) = \alpha  ^ {2} $
 +
for all $  \alpha \in F $.  
 +
It is $  \theta ( \alpha ) = \alpha ^ {2 ^ {m+ 1 } } $.
  
There is a twisted polarity whose absolute points are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151042.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151043.png" />-dimensional subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151044.png" /> that are fixed by a conjugate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151045.png" />. The set of absolute points thus obtained is an [[Ovoid(2)|ovoid]]. See [[#References|[a1]]]–[[#References|[a2]]].
+
There is a twisted polarity whose absolute points are the $  q  ^ {2} + 1 $
 +
$  1 $-
 +
dimensional subspaces of $  F _ {q}  ^ {q} $
 +
that are fixed by a conjugate of $  UT $.  
 +
The set of absolute points thus obtained is an [[Ovoid(2)|ovoid]]. See [[#References|[a1]]]–[[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Tits,  "Ovoïdes et groupes de Suzuki"  ''Arch. Math.'' , '''13'''  (1962)  pp. 187–198</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Tits,  "Une propriété charactéristique des ovoïdes associés aux groupes de Suzuki"  ''Arch. Math.'' , '''17'''  (1966)  pp. 136–153</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Huppert,  "Finite groups" , '''3''' , Springer  (1982)  pp. Chapt. IX.3</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Tits,  "Ovoïdes et groupes de Suzuki"  ''Arch. Math.'' , '''13'''  (1962)  pp. 187–198</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Tits,  "Une propriété charactéristique des ovoïdes associés aux groupes de Suzuki"  ''Arch. Math.'' , '''17'''  (1966)  pp. 136–153</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Huppert,  "Finite groups" , '''3''' , Springer  (1982)  pp. Chapt. IX.3</TD></TR></table>

Latest revision as of 14:55, 7 June 2020


A simple finite group $ \mathop{\rm Sz} ( q) $, a member of the infinite series of simple groups, discovered by M. Suzuki.

Let $ n $ be a natural number, let $ F $ be the finite field with $ q = 2 ^ {2n+} 1 $ elements, and let $ \theta $ be an automorphism of $ F $ such that $ \alpha ^ {\theta ^ {2} } = \alpha ^ {2} $ for any $ \alpha \in F $. The Suzuki group $ \mathop{\rm Sz} ( q) $ is then generated by the subgroup $ T $ consisting of all diagonal matrices of order 4 with diagonal elements $ \lambda ^ {1+ \theta } , \lambda , \lambda ^ {-} 1 , ( \lambda ^ {1+ \theta } ) ^ {-} 1 $( $ \lambda \in F $, $ \lambda \neq 0 $), the subgroup $ U $ consisting of all triangular matrices of the form

$$ \left \| \begin{array}{cccc} 1 & 0 & 0 & 0 \\ \alpha & 1 & 0 & 0 \\ \alpha ^ {1+ \theta } + \beta &\alpha ^ \theta & 1 & 0 \\ \alpha ^ {2+ \theta } + \alpha \beta + \beta ^ \theta &\beta &\alpha & 1 \\ \end{array} \right \| $$

$ ( \alpha , \beta \in F ) $, and the matrix

$$ \left \| \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right \| . $$

The subgroup $ U $ is a Sylow $ 2 $- subgroup of the group $ \mathop{\rm Sz} ( q) $; it is a Suzuki $ 2 $- group. The subgroup $ UT $ coincides with the normalizer of the subgroup $ U $. The permutation representation of the group $ \mathop{\rm Sz} ( q) $ on the cosets of $ UT $ is doubly transitive; its degree is equal to $ q ^ {2} + 1 $. The order of the Suzuki group $ \mathop{\rm Sz} ( q) $ is equal to $ q ^ {2} ( q- 1)( q ^ {2} + 1) $, and is not divisible by 3. On the other hand, any non-Abelian finite simple group whose order is not divisible by 3 is isomorphic to a Suzuki group. The group $ \mathop{\rm Sz} ( q) $ is a maximal subgroup of the symplectic group $ \mathop{\rm Sp} ( 4, q) $ and is the centralizer in $ \mathop{\rm Sp} ( 4, q) $ of an automorphism of order 2 of the group $ \mathop{\rm Sp} ( 4, q) = B _ {2} ( q) $. In other words, $ \mathop{\rm Sz} ( q) $ is isomorphic to $ {} ^ {2} B _ {2} ( q) $— the twisted analogue of a Chevalley group of type $ B _ {2} $ over the field with $ q $ elements.

References

[1] M. Suzuki, "On a class of doubly transitive groups" Ann. of Math. , 75 : 1 (1962) pp. 105–145
[2] R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972)

Comments

There is in fact precisely one automorphism $ \theta $ of $ F $ such that $ \theta ^ {2} ( \alpha ) = \alpha ^ {2} $ for all $ \alpha \in F $. It is $ \theta ( \alpha ) = \alpha ^ {2 ^ {m+ 1 } } $.

There is a twisted polarity whose absolute points are the $ q ^ {2} + 1 $ $ 1 $- dimensional subspaces of $ F _ {q} ^ {q} $ that are fixed by a conjugate of $ UT $. The set of absolute points thus obtained is an ovoid. See [a1][a2].

References

[a1] J. Tits, "Ovoïdes et groupes de Suzuki" Arch. Math. , 13 (1962) pp. 187–198
[a2] J. Tits, "Une propriété charactéristique des ovoïdes associés aux groupes de Suzuki" Arch. Math. , 17 (1966) pp. 136–153
[a3] B. Huppert, "Finite groups" , 3 , Springer (1982) pp. Chapt. IX.3
How to Cite This Entry:
Suzuki group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Suzuki_group&oldid=19048
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article