Difference between revisions of "Zassenhaus group"
From Encyclopedia of Mathematics
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| − | + | A doubly-transitive group $G$ of permutations on a finite set $M$ (cf. [[Permutation group]]) in which only the identity permutation fixes more than two elements of $M$ and such that for any pair $a,b \in M$ the subgroup $H_{a,b}$ is non-trivial, where | |
| − | + | $$ | |
| − | such groups were first considered by H. Zassenhaus in [[#References|[1]]]. The class of Zassenhaus groups includes two families of finite simple groups: the projective | + | H_{a,b} = \{ h \in G : h(a)=a\,,\ h(b)=b \} \ ; |
| + | $$ | ||
| + | such groups were first considered by H. Zassenhaus in [[#References|[1]]]. The class of Zassenhaus groups includes two families of finite simple groups: the projective [[special linear group]]s $\mathrm{PSL}(2,q)$, $q>3$, and the [[Suzuki group]]s. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Zassenhaus, "Kennzeichnung endlicher linearer Gruppen als Permutationsgruppen" ''Abh. Math. Sem. Univ. Hamburg'' , '''11''' (1935) pp. 17–40</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Gorenstein, "Finite groups" , Harper & Row (1968)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> H. Zassenhaus, "Kennzeichnung endlicher linearer Gruppen als Permutationsgruppen" ''Abh. Math. Sem. Univ. Hamburg'' , '''11''' (1935) pp. 17–40</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> D. Gorenstein, "Finite groups" , Harper & Row (1968)</TD></TR> | ||
| + | </table> | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Huppert, "Finite groups" , '''3''' , Springer (1967)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Huppert, "Finite groups" , '''3''' , Springer (1967)</TD></TR> | ||
| + | </table> | ||
Latest revision as of 19:15, 22 October 2017
2020 Mathematics Subject Classification: Primary: 20B20 [MSN][ZBL]
A doubly-transitive group $G$ of permutations on a finite set $M$ (cf. Permutation group) in which only the identity permutation fixes more than two elements of $M$ and such that for any pair $a,b \in M$ the subgroup $H_{a,b}$ is non-trivial, where $$ H_{a,b} = \{ h \in G : h(a)=a\,,\ h(b)=b \} \ ; $$ such groups were first considered by H. Zassenhaus in [1]. The class of Zassenhaus groups includes two families of finite simple groups: the projective special linear groups $\mathrm{PSL}(2,q)$, $q>3$, and the Suzuki groups.
References
| [1] | H. Zassenhaus, "Kennzeichnung endlicher linearer Gruppen als Permutationsgruppen" Abh. Math. Sem. Univ. Hamburg , 11 (1935) pp. 17–40 |
| [2] | D. Gorenstein, "Finite groups" , Harper & Row (1968) |
Comments
References
| [a1] | B. Huppert, "Finite groups" , 3 , Springer (1967) |
How to Cite This Entry:
Zassenhaus group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zassenhaus_group&oldid=42165
Zassenhaus group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zassenhaus_group&oldid=42165
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article