From Wikipedia, the free encyclopedia
In
mathematics , the Schreier refinement theorem of
group theory states that any two
subnormal series of
subgroups of a given group have equivalent refinements, where two series are equivalent if there is a
bijection between their
factor groups that sends each factor group to an
isomorphic one.
The theorem is named after the
Austrian
mathematician
Otto Schreier who proved it in 1928. It provides an elegant proof of the
Jordan–Hölder theorem . It is often proved using the
Zassenhaus lemma .
Baumslag (2006) gives a short proof by intersecting the terms in one subnormal series with those in the other series.
Example
Consider
Z
2
×
S
3
{\displaystyle \mathbb {Z} _{2}\times S_{3}}
, where
S
3
{\displaystyle S_{3}}
is the
symmetric group of degree 3 . The
alternating group
A
3
{\displaystyle A_{3}}
is a normal subgroup of
S
3
{\displaystyle S_{3}}
, so we have the two subnormal series
{
0
}
×
{
(
1
)
}
◃
Z
2
×
{
(
1
)
}
◃
Z
2
×
S
3
,
{\displaystyle \{0\}\times \{(1)\}\;\triangleleft \;\mathbb {Z} _{2}\times \{(1)\}\;\triangleleft \;\mathbb {Z} _{2}\times S_{3},}
{
0
}
×
{
(
1
)
}
◃
{
0
}
×
A
3
◃
Z
2
×
S
3
,
{\displaystyle \{0\}\times \{(1)\}\;\triangleleft \;\{0\}\times A_{3}\;\triangleleft \;\mathbb {Z} _{2}\times S_{3},}
with respective factor groups
(
Z
2
,
S
3
)
{\displaystyle (\mathbb {Z} _{2},S_{3})}
and
(
A
3
,
Z
2
×
Z
2
)
{\displaystyle (A_{3},\mathbb {Z} _{2}\times \mathbb {Z} _{2})}
.
The two subnormal series are not equivalent, but they have equivalent refinements:
{
0
}
×
{
(
1
)
}
◃
Z
2
×
{
(
1
)
}
◃
Z
2
×
A
3
◃
Z
2
×
S
3
{\displaystyle \{0\}\times \{(1)\}\;\triangleleft \;\mathbb {Z} _{2}\times \{(1)\}\;\triangleleft \;\mathbb {Z} _{2}\times A_{3}\;\triangleleft \;\mathbb {Z} _{2}\times S_{3}}
with factor groups isomorphic to
(
Z
2
,
A
3
,
Z
2
)
{\displaystyle (\mathbb {Z} _{2},A_{3},\mathbb {Z} _{2})}
and
{
0
}
×
{
(
1
)
}
◃
{
0
}
×
A
3
◃
{
0
}
×
S
3
◃
Z
2
×
S
3
{\displaystyle \{0\}\times \{(1)\}\;\triangleleft \;\{0\}\times A_{3}\;\triangleleft \;\{0\}\times S_{3}\;\triangleleft \;\mathbb {Z} _{2}\times S_{3}}
with factor groups isomorphic to
(
A
3
,
Z
2
,
Z
2
)
{\displaystyle (A_{3},\mathbb {Z} _{2},\mathbb {Z} _{2})}
.
References
Baumslag, Benjamin (2006), "A simple way of proving the Jordan-Hölder-Schreier theorem", American Mathematical Monthly , 113 (10): 933–935,
doi :
10.2307/27642092 ,
JSTOR
27642092