Let $G$ be a group, let $F$ be a normal subgroup of $G$, and let $E$ be a normal subgroup of $F$. That is, let $E$ be a normal subgroup of $F$, and let $F$ in turn be a normal subgroup of $G$. Is $E$ normal in $G$? If not necessarily, then what example illustrates this situation?
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asked Apr 7, 2014 at 2:49
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$\begingroup$ Please try to use search first. And here is the first hit I got googling this:groupprops.subwiki.org/wiki/Normality_is_not_transitive $\endgroup$– rschwiebApr 7, 2014 at 3:05
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$\begingroup$ Or, better still, try a bit harder to solve the problem yourself first. $\endgroup$– Derek HoltApr 7, 2014 at 8:29
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2 Answers
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Normality is not transitive. Let $A_4$ be the alternating group of 4th degree. Then the subgroup of double transpositions $V = \{ e, (12)(34),(13)(24),(14)(23)\}$ is a normal subgroup and if $Z = \{e,(12)(34)\}$ then $Z$ is a normal subgroup of $V$ that is not normal in $A_4$
answered Apr 7, 2014 at 3:03
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Counterexample:
$$\{(1),(12)(34)\}\lhd \{(1),(12)(34),(13)(24),(14)(23)\}\lhd A_4\;,\;\;\text{but}\;\;\{(1),(12)(34)\}\rlap{\;\,/}\lhd A_4$$
answered Apr 7, 2014 at 3:04