Difference between revisions of "Schreier's Theorem"
(New page: '''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 year...) |
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== Proof == | == Proof == | ||
| − | Suppose <math>\Sigma_1 = (H_i)_{0 \le i \le n | + | Suppose <math>\Sigma_1 = (H_i)_{0 \le i \le n}</math> and <math>\Sigma_2 = (K_j)_{0\le j \le m}</math> are the composition series in question. For integers <math>j \in [1,m-1]</math>, <math>i \in [0,n-1]</math>, let <math>H'_{im+j} = H_{i+1} \cdot (H_i \cap K_j)</math>, and for integers <math>i \in [0,n]</math>, let |
<cmath> H'_{im} = H_i = H_{i+1} \cdot (H_i \cap K_0) = H_{i} \cdot (H_{i-1} \cap K_m), </cmath> | <cmath> H'_{im} = H_i = H_{i+1} \cdot (H_i \cap K_0) = H_{i} \cdot (H_{i-1} \cap K_m), </cmath> | ||
where these groups are defined. Similarly, for integers <math>i \in [1,n-1]</math>, <math>j\in [0,m-1]</math>, let <math>K'_{jn+i} = K_{j+1} \cdot (K_j \cap H_i)</math>, and for integers <math>j \in [0,m]</math>, define | where these groups are defined. Similarly, for integers <math>i \in [1,n-1]</math>, <math>j\in [0,m-1]</math>, let <math>K'_{jn+i} = K_{j+1} \cdot (K_j \cap H_i)</math>, and for integers <math>j \in [0,m]</math>, define | ||
Revision as of 19:26, 21 January 2018
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 
and 
are the composition series in question. For integers ![$j \in [1,m-1]$](http://latex.artofproblemsolving.com/3/9/3/3937af7ad96312652a97b7f9a2a8c8d6a8d34963.png)
, ![$i \in [0,n-1]$](http://latex.artofproblemsolving.com/6/f/4/6f459a163e3ae43553cbefbd1f0ac5b3160e7e92.png)
, let 
, and for integers ![$i \in [0,n]$](http://latex.artofproblemsolving.com/1/f/e/1fec662283227219e749fd72a1269cc04d7fdbc0.png)
, let
![\[H'_{im} = H_i = H_{i+1} \cdot (H_i \cap K_0) = H_{i} \cdot (H_{i-1} \cap K_m),\]](http://latex.artofproblemsolving.com/e/e/9/ee92e6cf7ddbcbe68289aae8e6a4aeb879c7a253.png)
where these groups are defined. Similarly, for integers ![$i \in [1,n-1]$](http://latex.artofproblemsolving.com/6/4/e/64ee1a7568a0cd9332a713ff47076a77c671007c.png)
, ![$j\in [0,m-1]$](http://latex.artofproblemsolving.com/0/1/f/01f164dea13b552a7d664675a1fdec80fcaadbf4.png)
, let 
, and for integers ![$j \in [0,m]$](http://latex.artofproblemsolving.com/5/b/8/5b8e062eb58426db4e5498947145523fa8340e02.png)
, define
![\[K'_{jn} = K_j = K_{j+1} \cdot (K_j \cap H_0) = K_j \cdot (K_{j-1} \cap H_n),\]](http://latex.artofproblemsolving.com/2/d/7/2d703bc763d41b14a0dfa680010b6f3a5e48cb9d.png)
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. 
