# Zassenhaus group

2010 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)