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Difference between revisions of "Suzuki-2-group"

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<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>
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<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>
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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=32393
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article