Difference between revisions of "Schreier's Theorem"
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Revision as of 22:15, 10 May 2008
Schreier's Refinement Theorem is a result in group theory. Otto Schreir discovered it in 1928, and used it to give an improved proof of the Jordan-Hölder Theorem. Six years later, Hans Zassenhaus published his lemma, which gives an improved proof of Schreier's Theorem.
Statement
Let and
be composition series of a group
. Then there exist equivalent composition series
and
such that
is finer than
and
is finer than
.
Proof
Suppose $\Sigma_1 = (H_i)_{0 \le i \le n)$ (Error compiling LaTeX. Unknown error_msg) and are the composition series in question. For integers
,
, let
, and for integers
, let
where these groups are defined. Similarly, for integers
,
, let
, and for integers
, define
where these groups are defined. Then by Zassenhaus's Lemma,
and
are composition series; they are evidently finer than
and
, respectively. Again by Zassenhaus's Lemma, the quotients
and
are equivalent, so series
and
are equivalent, as desired.