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For a subgroup $H$ of the finite group $G$, define $C(H)$ to be the set of all subgroups of $G$ which are permutable with $H$, i.e. $C(H) \:= \{K \leq G: HK=KH \}$.

My question is: can the class of groups $G$ with the property that $C(H) \:= \{K \leq N_G(H): K \leq G \} \cup N(G)$ for all $H \leq G$, where $N(G)$ is the set of normal subgroups of $G$, be described in some way?

For example, Dedekind groups, that is groups all of whose subgroups are normal, belong to this class and their structure is known.

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As you have defined it, $G$ will always be in $C(H).$ Therefore, unless you exclude $G$, you will get $G \leq N_G(H)$ for each subgroup $H$ of $G$ for groups in your class, so $G$ will be a Dedekind group. –  Geoff Robinson Mar 11 '12 at 8:41
    
Thanks, I've made a couple of corrections. –  the_fox Mar 11 '12 at 9:30
    
If I understand what you are writing correctly, you are asking that $G$ satisfy the following property: if $HK=KH$, then either $H\subseteq N_G(K)$, or $K\subseteq N_G(H)$. Is this correct? –  Arturo Magidin Mar 11 '12 at 22:17
    
If my reading is correct, I know I've seen papers discussing this property, but I'm having a hard time tracking them down. More if I succeed. –  Arturo Magidin Mar 11 '12 at 22:30
    
@Kannappan: Actually, the terminology was correct: in Group Theory, you say subgroups $H$ and $K$ are "permutable" if $HK=KH$. –  Arturo Magidin Mar 11 '12 at 22:36
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