Schreier refinement theorem

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 ${\displaystyle \mathbb {Z} /(2)\times S_{3}}$, where ${\displaystyle S_{3}}$ is the symmetric group of degree 3. There are subnormal series

${\displaystyle \{[0]\}\times \{\operatorname {id} \}\;\triangleleft \;\mathbb {Z} /(2)\times \{\operatorname {id} \}\;\triangleleft \;\mathbb {Z} /(2)\times S_{3},}$

${\displaystyle \{[0]\}\times \{\operatorname {id} \}\;\triangleleft \;\{[0]\}\times S_{3}\;\triangleleft \;\mathbb {Z} /(2)\times S_{3}.}$

${\displaystyle S_{3}}$ contains the normal subgroup ${\displaystyle A_{3}}$. Hence these have refinements

${\displaystyle \{[0]\}\times \{\operatorname {id} \}\;\triangleleft \;\mathbb {Z} /(2)\times \{\operatorname {id} \}\;\triangleleft \;\mathbb {Z} /(2)\times A_{3}\;\triangleleft \;\mathbb {Z} /(2)\times S_{3}}$

with factor groups isomorphic to ${\displaystyle (\mathbb {Z} /(2),A_{3},\mathbb {Z} /(2))}$ and

${\displaystyle \{[0]\}\times \{\operatorname {id} \}\;\triangleleft \;\{[0]\}\times A_{3}\;\triangleleft \;\{[0]\}\times S_{3}\;\triangleleft \;\mathbb {Z} /(2)\times S_{3}}$

with factor groups isomorphic to ${\displaystyle (A_{3},\mathbb {Z} /(2),\mathbb {Z} /(2))}$.

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