# Schreier refinement theorem

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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.

## Example

Consider Z / ( 2 ) × S 3 , where S 3 is the symmetric group of degree 3. There are subnormal series

{ [ 0 ] } × { id } Z / ( 2 ) × { id } Z / ( 2 ) × S 3 , { [ 0 ] } × { id } { [ 0 ] } × S 3 Z / ( 2 ) × S 3 .

S 3 contains the normal subgroup A 3 . Hence these have refinements

{ [ 0 ] } × { id } Z / ( 2 ) × { id } Z / ( 2 ) × A 3 Z / ( 2 ) × S 3

with factor groups isomorphic to ( Z / ( 2 ) , A 3 , Z / ( 2 ) ) and

{ [ 0 ] } × { id } { [ 0 ] } × A 3 { [ 0 ] } × S 3 Z / ( 2 ) × S 3

with factor groups isomorphic to ( A 3 , Z / ( 2 ) , Z / ( 2 ) ) .

## References

Schreier refinement theorem Wikipedia

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