Properties of a group

Some properties of a group $G$

Let $K\lhd G$, $K\subset H$ be a subgroup of $G$.

1

Map
\begin{align}
\begin{array}{ccc}
\{H\,|\,K\subset H:\text{subgroup of } G\}
&\rightarrow
&\{\bar{H}\,|\bar{H}:\text{subgroup of } G/K\}\\
H&\mapsto&H/K
\end{array}
\end{align}
is bijective.

• Proof. Let $\bar{H}= \{aK\,|\,a\in A\}$. Then $\displaystyle\bigcup_{a\in A}aK\in G$ and $\displaystyle\bar{H}= \{aK\,|\,a\in A\}\mapsto \bigcup_{a\in A}aK$ is an inverse.

2

\begin{align}
(K\subset)H\lhd G
\Leftrightarrow
H/K\lhd G/K
\end{align}

• Proof. Easily shown from the definition.

3

If $(K\subset)H\lhd G$
\begin{align}
G/H \cong (G/K)\bigl/(H/K)
\end{align}

• Proof. Kernel of the homomorphism \begin{align}\begin{array}{cccc}G&\rightarrow&G/K&\rightarrow&(G/K)\bigl/(H/K)\\g&\mapsto&gK&\mapsto& gK(H/K)\\&&&\end{array}\end{align} is $H$.

Now, let $f:G\rightarrow G^\prime$ be epimorphism. Then we can get following generalised results(*1) by isomorphism $G/\mathrm{Ker\,}f\ni g\mathrm{Ker\,}f\mapsto f(g)\in G^\prime$.

1'

Map
\begin{align}
\begin{array}{ccc}
\{H\,|\,\mathrm{Ker\,}f\subset H:\text{subgroup of } G\}
&\rightarrow
&\{H^\prime\,|H^\prime:\text{subgroup of } G^\prime\}\\
H&\mapsto&f(H)
\end{array}
\end{align}
is bijective.

2'

\begin{align}
(\mathrm{Ker\,}f\subset)H\lhd G
\Leftrightarrow
f(H)\lhd G^\prime
\end{align}

3'

If $(\mathrm{Ker\,}f\subset)H\lhd G$
\begin{align}
G/H \cong G^\prime/f(H)
\end{align}

*1:Consider the case $f:G\ni g\mapsto gK\in G/K$.

How to rename files using script

Shell script

Extract from 3rd to 5th characters of the file name

#!/bin/sh
dir=Path of the working directory
cd $dir
for file in *.txt
do
    After=`echo $file | cut -c 3-5`
    mv "${file}" "${After}.txt"
done

#!/bin/sh
dir=Path of the working directory
cd $dir
for file in *.txt
do
    After="${file:2:3}"
    extension="${file##*.}"
    mv "${file}" "${After}.${extension}"
done

Add prefix "P-" and suffix "-S"

#!/bin/sh
Prefix=P-
Suffix=-S
dir=Path of the working directory
cd $dir
for file in *.txt
do
    filename="${file%.*}"
    extension="${file##*.}"
    mv "${file}" "${Prefix}${filename}${Suffix}.${extension}"
done

Batch file (For MS DOS)

Sometimes, we cannot use shell commands.
In such a case, Batch file for windows may be useful.

To execute following programs,
copy and paste code to "Notepad" and save with ".bat" extension.

Ex. : Delete first two characters. (abcde.txt→cde.txt)

Edit file name except for extension.

for %%F in (*.txt) do call :sub "%%F"
goto :EOF

:sub
  set BEFORE=%~1
  set FILENAME=%~n1
  set EXTENSION=%~x1
  set AFTER=%FILENAME:~3%%EXTENSION%
  ren "%BEFORE%" "%AFTER%"
goto :EOF

• %~x1: Get Extension of the file
• %~n1: Get filename without extension
• ren A B: Rename filename from A to B

Ex. : Add "prefix" at the beginning of a file name. (abc.txt→prefixabc.txt)

To add character string "prefix" to the beginning of the filename,
write just like this.

for %%F in (*.txt) do ren "%%F" "prefix%%F"

Ex. : Add "suffix" at the end of the file name. (abc.txt→abcsuffix.txt)

Same idea as above, but be careful not to edit extension.
To do this, decompose filename before editing:

Filename + suffix + Extension

for %%F in (*.txt) do ren "%%F" "%%~nFsuffix%%~xF"

Determinant of ...

Determinant of $r^2\mathbb{1}-3\boldsymbol{r}\otimes\boldsymbol{r}$

\begin{align}
\boldsymbol{r}\otimes\boldsymbol{r}
=\begin{pmatrix}
x^2&xy&xz\\
yx&y^2&yz\\
zx&zy&z^2
\end{pmatrix}
\end{align}
This matrix has following properties:

1. $\mathrm{rank}\,\boldsymbol{r}\otimes\boldsymbol{r}=1$
2. $\mathrm{Tr}\,\boldsymbol{r}\otimes\boldsymbol{r}=r^2$

Therefore, we can change basis such that*1
\begin{align}
P^{-1}(\boldsymbol{r}\otimes\boldsymbol{r})P
=\begin{pmatrix}
r^2&0&0\\
*&0&0\\
*&0&0
\end{pmatrix}
\end{align}

Consequently (Note: $\det A=\det (P^{-1}AP)$),
\begin{align}
\det \left(r^2\mathbb{1}-3\boldsymbol{r}\otimes\boldsymbol{r}\right)
=-2r^6.
\end{align}

*1:Chose one from $\mathrm{Im}\,(\boldsymbol{r}\otimes\boldsymbol{r})$ and the others from $\mathrm{Ker}\,(\boldsymbol{r}\otimes\boldsymbol{r})$