Editing Schreier refinement theorem (section)

== Example ==

Consider <math>\mathbb{Z}_2 \times S_3</math>, where <math>S_3</math> is the [[symmetric group of degree 3]].  The [[alternating group]] <math>A_3</math> is a normal subgroup of <math>S_3</math>, so we have the two subnormal series
<div style="text-align: center;">
: <math>\{0\} \times \{(1)\} \; \triangleleft \; \mathbb{Z}_2 \times \{(1)\} \; \triangleleft \; \mathbb{Z}_2 \times S_3,</math>
: <math>\{0\} \times \{(1)\} \; \triangleleft \; \{0\} \times A_3 \; \triangleleft \; \mathbb{Z}_2 \times S_3,</math>
</div>
with respective factor groups <math>(\mathbb{Z}_2,S_3)</math> and <math>(A_3,\mathbb{Z}_2\times\mathbb{Z}_2)</math>.<br>
The two subnormal series are not equivalent, but they have equivalent refinements:

: <math>\{0\} \times \{(1)\} \; \triangleleft \; \mathbb{Z}_2 \times \{(1)\} \; \triangleleft \; \mathbb{Z}_2 \times A_3 \; \triangleleft \; \mathbb{Z}_2 \times S_3</math>

with factor groups isomorphic to <math>(\mathbb{Z}_2, A_3, \mathbb{Z}_2)</math> and

: <math>\{0\} \times \{(1)\} \; \triangleleft \; \{0\} \times A_3 \; \triangleleft \; \{0\} \times S_3 \; \triangleleft \; \mathbb{Z}_2 \times S_3</math>

with factor groups isomorphic to <math>(A_3, \mathbb{Z}_2, \mathbb{Z}_2)</math>.