View source for Suzuki group

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.

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

<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>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091510/s09151016.png" />, and the matrix

<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>

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.

====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>



====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 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]]].

====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>