<s>
Suppose	O
G	O
is	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
consisting	O
of	O
permutations	O
of	O
a	O
set	O
X	O
.	O
</s>
<s>
A	O
subgroup	O
H	O
of	O
G	O
fixing	O
a	O
point	O
of	O
X	O
is	O
called	O
a	O
Frobenius	B-Algorithm
complement	I-Algorithm
.	O
</s>
<s>
The	O
identity	O
element	O
together	O
with	O
all	O
elements	O
not	O
in	O
any	O
conjugate	O
of	O
H	O
form	O
a	O
normal	O
subgroup	O
called	O
the	O
Frobenius	B-Algorithm
kernel	I-Algorithm
K	O
.	O
(	O
This	O
is	O
a	O
theorem	O
due	O
to	O
;	O
there	O
is	O
still	O
no	O
proof	O
of	O
this	O
theorem	O
that	O
does	O
not	O
use	O
character	O
theory	O
,	O
although	O
see	O
.	O
)	O
</s>
<s>
The	O
Frobenius	B-Algorithm
group	I-Algorithm
G	O
is	O
the	O
semidirect	O
product	O
of	O
K	O
and	O
H	O
:	O
</s>
<s>
Both	O
the	O
Frobenius	B-Algorithm
kernel	I-Algorithm
and	O
the	O
Frobenius	B-Algorithm
complement	I-Algorithm
have	O
very	O
restricted	O
structures	O
.	O
</s>
<s>
proved	O
that	O
the	O
Frobenius	B-Algorithm
kernel	I-Algorithm
K	O
is	O
a	O
nilpotent	O
group	O
.	O
</s>
<s>
The	O
Frobenius	B-Algorithm
complement	I-Algorithm
H	O
has	O
the	O
property	O
that	O
every	O
subgroup	O
whose	O
order	O
is	O
the	O
product	O
of	O
2	O
primes	O
is	O
cyclic	O
;	O
this	O
implies	O
that	O
its	O
Sylow	O
subgroups	O
are	O
cyclic	O
or	O
generalized	O
quaternion	O
groups	O
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
a	O
Frobenius	B-Algorithm
complement	I-Algorithm
coincides	O
with	O
its	O
derived	O
subgroup	O
,	O
then	O
it	O
is	O
isomorphic	O
with	O
SL(2,5 )	O
.	O
</s>
<s>
If	O
a	O
Frobenius	B-Algorithm
complement	I-Algorithm
H	O
is	O
solvable	O
then	O
it	O
has	O
a	O
normal	O
metacyclic	O
subgroup	O
such	O
that	O
the	O
quotient	O
is	O
a	O
subgroup	O
of	O
the	O
symmetric	O
group	O
on	O
4	O
points	O
.	O
</s>
<s>
A	O
finite	O
group	O
is	O
a	O
Frobenius	B-Algorithm
complement	I-Algorithm
if	O
and	O
only	O
if	O
it	O
has	O
a	O
faithful	O
,	O
finite-dimensional	O
representation	O
over	O
a	O
finite	O
field	O
in	O
which	O
non-identity	O
group	O
elements	O
correspond	O
to	O
linear	O
transformations	O
without	O
nonzero	O
fixed	O
points	O
.	O
</s>
<s>
The	O
Frobenius	B-Algorithm
kernel	I-Algorithm
K	O
is	O
uniquely	O
determined	O
by	O
G	O
as	O
it	O
is	O
the	O
Fitting	O
subgroup	O
,	O
and	O
the	O
Frobenius	B-Algorithm
complement	I-Algorithm
is	O
uniquely	O
determined	O
up	O
to	O
conjugacy	O
by	O
the	O
Schur-Zassenhaus	O
theorem	O
.	O
</s>
<s>
In	O
particular	O
a	O
finite	O
group	O
G	O
is	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
in	O
at	O
most	O
one	O
way	O
.	O
</s>
<s>
The	O
Frobenius	B-Algorithm
kernel	I-Algorithm
K	O
has	O
order	O
3	O
,	O
and	O
the	O
complement	O
H	O
has	O
order	O
2	O
.	O
</s>
<s>
For	O
every	O
finite	O
field	O
Fq	O
with	O
q	O
(	O
2	O
)	O
elements	O
,	O
the	O
group	O
of	O
invertible	O
affine	B-Algorithm
transformations	I-Algorithm
,	O
acting	O
naturally	O
on	O
Fq	O
is	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
This	O
Frobenius	B-Algorithm
group	I-Algorithm
acts	O
simply	O
transitively	O
on	O
the	O
21	O
flags	O
in	O
the	O
Fano	O
plane	O
,	O
i.e.	O
</s>
<s>
The	O
dihedral	B-Algorithm
group	I-Algorithm
of	O
order	O
2n	O
with	O
n	O
odd	O
is	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
with	O
complement	O
of	O
order	O
2	O
.	O
</s>
<s>
More	O
generally	O
if	O
K	O
is	O
any	O
abelian	O
group	O
of	O
odd	O
order	O
and	O
H	O
has	O
order	O
2	O
and	O
acts	O
on	O
K	O
by	O
inversion	O
,	O
then	O
the	O
semidirect	O
product	O
K.H	O
is	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
If	O
we	O
replace	O
the	O
Frobenius	B-Algorithm
complement	I-Algorithm
of	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
by	O
a	O
non-trivial	O
subgroup	O
we	O
get	O
another	O
Frobenius	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
If	O
we	O
have	O
two	O
Frobenius	B-Algorithm
groups	I-Algorithm
K1.H	O
and	O
K2.H	O
then	O
(	O
K1K2	O
)	O
.H	O
is	O
also	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
If	O
K	O
is	O
the	O
non-abelian	O
group	O
of	O
order	O
73	O
with	O
exponent	O
7	O
,	O
and	O
H	O
is	O
the	O
cyclic	O
group	O
of	O
order	O
3	O
,	O
then	O
there	O
is	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
G	O
that	O
is	O
an	O
extension	O
K.H	O
of	O
H	O
by	O
K	O
.	O
This	O
gives	O
an	O
example	O
of	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
with	O
non-abelian	O
kernel	O
.	O
</s>
<s>
This	O
was	O
the	O
first	O
example	O
of	O
Frobenius	B-Algorithm
group	I-Algorithm
with	O
nonabelian	O
kernel	O
(	O
it	O
was	O
constructed	O
by	O
Otto	O
Schmidt	O
)	O
.	O
</s>
<s>
The	O
extension	O
K.H	O
is	O
the	O
smallest	O
example	O
of	O
a	O
non-solvable	O
Frobenius	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
subgroup	O
of	O
a	O
Zassenhaus	B-Algorithm
group	I-Algorithm
fixing	O
a	O
point	O
is	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Frobenius	B-Algorithm
groups	I-Algorithm
whose	O
Fitting	O
subgroup	O
has	O
arbitrarily	O
large	O
nilpotency	O
class	O
were	O
constructed	O
by	O
Ito	O
:	O
Let	O
q	O
be	O
a	O
prime	O
power	O
,	O
d	O
a	O
positive	O
integer	O
,	O
and	O
p	O
a	O
prime	O
divisor	O
of	O
q	O
1	O
with	O
d	O
≤	O
p	O
.	O
Fix	O
some	O
field	O
F	O
of	O
order	O
q	O
and	O
some	O
element	O
z	O
of	O
this	O
field	O
of	O
order	O
p	O
.	O
The	O
Frobenius	B-Algorithm
complement	I-Algorithm
H	O
is	O
the	O
cyclic	O
subgroup	O
generated	O
by	O
the	O
diagonal	O
matrix	O
whose	O
i	O
,	O
i'''th	O
entry	O
is	O
zi	O
.	O
</s>
<s>
The	O
Frobenius	B-Algorithm
kernel	I-Algorithm
K	O
is	O
the	O
Sylow	O
q-subgroup	O
of	O
GL(d,q )	O
consisting	O
of	O
upper	O
triangular	O
matrices	O
with	O
ones	O
on	O
the	O
diagonal	O
.	O
</s>
<s>
The	O
kernel	O
K	O
has	O
nilpotency	O
class	O
d	O
−1	O
,	O
and	O
the	O
semidirect	O
product	O
KH	O
is	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
irreducible	O
complex	O
representations	O
of	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
G	O
can	O
be	O
read	O
off	O
from	O
those	O
of	O
H	O
and	O
K	O
.	O
There	O
are	O
two	O
types	O
of	O
irreducible	O
representations	O
of	O
G	O
:	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
group	O
theoretical	O
properties	O
which	O
are	O
interesting	O
on	O
their	O
own	O
right	O
,	O
but	O
which	O
happen	O
to	O
be	O
equivalent	O
to	O
the	O
group	O
possessing	O
a	O
permutation	O
representation	O
that	O
makes	O
it	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
G	O
is	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
if	O
and	O
only	O
if	O
G	O
has	O
a	O
proper	O
,	O
nonidentity	O
subgroup	O
H	O
such	O
that	O
H	O
∩	O
Hg	O
is	O
the	O
identity	O
subgroup	O
for	O
every	O
g	O
∈	O
G	O
−	O
H	O
,	O
i.e.	O
</s>
<s>
This	O
definition	O
is	O
then	O
generalized	O
to	O
the	O
study	O
of	O
trivial	O
intersection	O
sets	O
which	O
allowed	O
the	O
results	O
on	O
Frobenius	B-Algorithm
groups	I-Algorithm
used	O
in	O
the	O
classification	O
of	O
CA	O
groups	O
to	O
be	O
extended	O
to	O
the	O
results	O
on	O
CN	O
groups	O
and	O
finally	O
the	O
odd	O
order	O
theorem	O
.	O
</s>
<s>
Assuming	O
that	O
is	O
the	O
semidirect	O
product	O
of	O
the	O
normal	O
subgroup	O
K	O
and	O
complement	O
H	O
,	O
then	O
the	O
following	O
restrictions	O
on	O
centralizers	O
are	O
equivalent	O
to	O
G	O
being	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
with	O
Frobenius	B-Algorithm
complement	I-Algorithm
H	O
:	O
</s>
