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Given the projection $\pi_A$ from a group $G$ to $G/A$ where $A$ is normal, is there a name and/or a standard notation for $\pi_A^{-1}\left(Z\left(G/A\right)\right)$?

I came across this object in my studies of racks and I wondered if it had been already named in the literature but I couldn't find any easy reference.

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Without a name, the notation $Z(G mod A)$ is used by A. Mann in "Elements of minimal breadth...", J. Aust. Math. Soc. 81 (2006). –  Yassine Guerboussa Jul 3 '13 at 15:02
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I don't think it has a special name, except in the special case where $A=Z(G)$ (in which case you have the "second center", $Z_2(G)$), or more generally, obtained by such an iteration, producing the upper central series of $G$, $Z(G)=Z_1(G)\leq Z_2(G)\leq Z_3(G)\leq\cdots$. –  Arturo Magidin Jul 3 '13 at 16:32
    
I think this is the occasion to ask specialists to suggest a nice name and notation. –  Yassine Guerboussa Jul 3 '13 at 17:39
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How about "precenter"? –  Ian Agol Jul 4 '13 at 0:28
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@Giuliano: I doubt that a name exists, since the concept depends not just on $G$ but also on which normal subgroup $A$ you work with. (Small side note: a letter like $H$ or $N$ might be better, not suggesting "abelian" as the letter $A$ does. Unless that special case is what you have in mind.) –  Jim Humphreys Jul 4 '13 at 0:29

1 Answer 1

Since, there seem not to be an accepted term, I hereby propose, accepting the suggestion of Prof. Agol

Definition. Given a surjective morphism of groups $f:G \rightarrow H $ the preimage of the center of $H$ shall be the precenter of $f$ and shall be indicated by $Z^{-1} \left( f \right)$.

The case proposed in my question would became $ Z^{-1}\left( \pi_A \right)$.

The jury is out :-)

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To my mind, it is better to find a notation that does not refer to any homomorphism, only to the subgroup and the whole group. –  Yassine Guerboussa Jul 4 '13 at 18:07
    
@YassineGuerboussa Make your proposal then! The sense of mine was to be somewhat categorical, the morphisms and not the objects are the heroes... –  Giuliano Bianco Jul 4 '13 at 18:12
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I suggest $Z_G(N)$, $N$ normal in $G$, and I would call it the center of $G$ modulo $N$. The precenter of $f$ is the center of $G$ modulo $ker(f)$, for brevity $Z_G(f)$ (or $Z(f)$). (Is it useful to see Group Theory from a categorical aspect?). Sincerely. –  Yassine Guerboussa Jul 4 '13 at 19:31
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That's the notation for centralizer of N in G. –  plusepsilon.de Sep 3 '13 at 5:06

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