# Zassenhaus group

From Encyclopedia of Mathematics

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

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