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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" />. | + | {{TEX|done}} |
| | + | 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 $q^3$. |
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| − | 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. | + | 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. |
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| | ====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 {{ZBL|0112.02107}}</TD></TR> |
| | + | </table> |
Latest revision as of 06:29, 19 May 2024
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 Zbl 0112.02107 |
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