Is it true that Subgroup of a Cyclic Normal subgroup of a Group is again Normal ? If so any hints for the proof?
A subgroup of a cyclic normal subgroup of a Group is Normal
1
$\begingroup$
abstract-algebra
group-theory
3 Answers
3
Sure. It follows from a more general fact: a characteristic subgroup of a normal subgroup of $G$ is also a normal subgroup of $G$. It's even easier to think about the question in these general terms.
2
If $H
-
0But how do we prove $H$ is a _characteristic subgroup_ of $N$ ? – 2012-11-22
0
Yes. This is one of the several cases when normality is transitive. If
$N\triangleleft C\triangleleft G\,\,\,,\,\,C=\text{cyclic, then}\;\; C=\langle c\rangle\Longrightarrow N=\langle c^k\rangle\Longrightarrow$
$\Longrightarrow \,\,\forall g\in G\,\,,\,\,x^{-1}c^{rk}x=(x^{-1}c^kx)^r\in N\,....$
-
0I believe it would have been clearer to use $(x^{-1}c^rx)^k \in N $ in the last line. – 2014-03-01