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.
Consider , where is the symmetric group of degree 3. The alternating group is a normal subgroup of , so we have the two subnormal series
,
with respective factor groups and . The two subnormal series are not equivalent, but they have equivalent refinements:
with factor groups isomorphic to and
with factor groups isomorphic to .
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, JSTOR27642092
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