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A finite non-Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s0915002.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s0915003.png" />, other than the group of quaternions, which admits a cyclic group of automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s0915004.png" /> that acts transitively on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s0915005.png" /> of elements of order 2 of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s0915006.png" />. This means that for any two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s0915007.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s0915008.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s0915009.png" /> there is a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150011.png" />. In the Suzuki <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150012.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150013.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150014.png" /> and the identity element constitute a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150015.png" /> that coincides with the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150016.png" />; the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150017.png" /> is then elementary Abelian. If the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150018.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150019.png" />, then the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150020.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150021.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150022.png" />.
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A finite non-Abelian $2$-group $U$, other than the group of quaternions, which admits a cyclic group of automorphisms $\langle a\rangle$ that acts transitively on the set $\Omega$ of elements of order 2 of $U$. This means that for any two elements $x$ and $y$ of $\Omega$ there is a natural number $n$ such that $y=x^{a^n}$. In the Suzuki $2$-group $U$, the set $\Omega$ and the identity element constitute a subgroup $Z$ that coincides with the centre of $U$; the quotient group $U/Z$ is then elementary Abelian. If the order of $Z$ is equal to $q$, then the order of $U$ is equal to $q^2$ or $q^3$.
  
Suzuki <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150023.png" />-groups have been fully described (see [[#References|[1]]]). The name derives from the fact that in a [[Suzuki group|Suzuki group]], the Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150024.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091500/s09150025.png" /> has these properties.
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Suzuki $2$-groups have been fully described (see [[#References|[1]]]). The name derives from the fact that in a [[Suzuki group|Suzuki group]], the Sylow $2$-group $U$ has these properties.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Higman,  "Suzuki 2-groups"  ''Ill. J. Math.'' , '''7''' :  1  (1963)  pp. 79–96</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Higman,  "Suzuki 2-groups"  ''Ill. J. Math.'' , '''7''' :  1  (1963)  pp. 79–96</TD></TR></table>

Revision as of 19:34, 7 July 2014

A finite non-Abelian $2$-group $U$, other than the group of quaternions, which admits a cyclic group of automorphisms $\langle a\rangle$ that acts transitively on the set $\Omega$ of elements of order 2 of $U$. This means that for any two elements $x$ and $y$ of $\Omega$ there is a natural number $n$ such that $y=x^{a^n}$. In the Suzuki $2$-group $U$, the set $\Omega$ and the identity element constitute a subgroup $Z$ that coincides with the centre of $U$; the quotient group $U/Z$ is then elementary Abelian. If the order of $Z$ is equal to $q$, then the order of $U$ is equal to $q^2$ or $q^3$.

Suzuki $2$-groups have been fully described (see [1]). The name derives from the fact that in a Suzuki group, the Sylow $2$-group $U$ has these properties.

References

[1] G. Higman, "Suzuki 2-groups" Ill. J. Math. , 7 : 1 (1963) pp. 79–96
How to Cite This Entry:
Suzuki-2-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Suzuki-2-group&oldid=13964
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article