<s>
In	O
mathematics	O
,	O
a	O
Zassenhaus	B-Algorithm
group	I-Algorithm
,	O
named	O
after	O
Hans	O
Zassenhaus	O
,	O
is	O
a	O
certain	O
sort	O
of	O
doubly	B-Algorithm
transitive	I-Algorithm
permutation	I-Algorithm
group	I-Algorithm
very	O
closely	O
related	O
to	O
rank-1	O
groups	O
of	O
Lie	O
type	O
.	O
</s>
<s>
A	O
Zassenhaus	B-Algorithm
group	I-Algorithm
is	O
a	O
permutation	O
group	O
G	O
on	O
a	O
finite	O
set	O
X	O
with	O
the	O
following	O
three	O
properties	O
:	O
</s>
<s>
G	O
is	O
doubly	B-Algorithm
transitive	I-Algorithm
.	O
</s>
<s>
The	O
degree	O
of	O
a	O
Zassenhaus	B-Algorithm
group	I-Algorithm
is	O
the	O
number	O
of	O
elements	O
of	O
X	O
.	O
</s>
<s>
2p(2p1 )	O
p	O
for	O
a	O
prime	O
p	O
,	O
that	O
are	O
generated	O
by	O
all	O
semilinear	B-Algorithm
mappings	I-Algorithm
and	O
Galois	O
automorphisms	O
of	O
a	O
field	O
of	O
order	O
2p	O
.	O
</s>
<s>
We	O
let	O
q	O
=	O
pf	O
be	O
a	O
power	O
of	O
a	O
prime	O
p	O
,	O
and	O
write	O
Fq	O
for	O
the	O
finite	O
field	O
of	O
order	O
q	O
.	O
Suzuki	O
proved	O
that	O
any	O
Zassenhaus	B-Algorithm
group	I-Algorithm
is	O
of	O
one	O
of	O
the	O
following	O
four	O
types	O
:	O
</s>
