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The normal subgroups of a group such that it is cyclic
In every group $G$ it is possible to find a subgroup that is normal in $G$ and cyclic:
$$\left \{ \begin{array}{rl}
G &= \langle g \rangle \\
H &= \langle g \rangle = \{ g^1, g^2,…, g^n,..\}
\end{array} \right.$$
I want to know what are the structure of the cyclic subgroups that are normal in $G$
A:
For a cyclic group, every group automorphism is of the form $g\mapsto kg$ for some $k\in\mathbb{Z}$. The eigenvalues of these groups act on the group through permutation of the elements. The resulting cyclic subgroups are all the possible subgroups of order $n$ where $n$ is the order of the given group.
Q:
Bad substitution inside bash string substitution with nested variable
Consider the following bash script:
#!/bin/bash
user=$1
# ignore user@host overrides
# if $var has value ‘root@host1’, then set $user to root
if [[ “$var” == “root@host1” ]]; then
eval ‘user=root’
fi
# resolve user@host overrides
# if $var has value ‘user@host1’, then set $user to ‘user@host1’
if [[ “$var” == “user@host1” ]]; then
eval ‘user=”$var”‘
fi
echo $user
I want to assign some user, e.g. root, and if a user override exists, e.g. user@host1, I want to assign it to the user variable, e.g. change $user to root@host1.
This script will incorrectly assign $user to root@host1 (see comments in script).
How can I resolve this issue?
A:
The following fixes
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