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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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    $\begingroup$ 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). $\endgroup$
    – Yassine Guerboussa
    Commented Jul 3, 2013 at 15:02
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    $\begingroup$ 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 $G$, $Z(G)=Z_1(G)\leq Z_2(G)\leq Z_3(G)\leq\cdots$. $\endgroup$
    – Arturo Magidin
    Commented Jul 3, 2013 at 16:32
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    $\begingroup$ How about "precenter"? $\endgroup$
    – Ian Agol
    Commented Jul 4, 2013 at 0:28
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    $\begingroup$ @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.) $\endgroup$
    – Jim Humphreys
    Commented Jul 4, 2013 at 0:29
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    $\begingroup$ @ Jim Humphreys : There is some concepts depending on G and a subgroup, which in fact have names, for instance "centralizer" and "normalizer". $\endgroup$
    – Yassine Guerboussa
    Commented Jul 4, 2013 at 11:04

2 Answers 2

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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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  • $\begingroup$ 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. $\endgroup$
    – Yassine Guerboussa
    Commented Jul 4, 2013 at 18:07
  • $\begingroup$ @YassineGuerboussa Make your proposal then! The sense of mine was to be somewhat categorical, the morphisms and not the objects are the heroes... $\endgroup$
    – Giuliano Bianco
    Commented Jul 4, 2013 at 18:12
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    $\begingroup$ 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. $\endgroup$
    – Yassine Guerboussa
    Commented Jul 4, 2013 at 19:31
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    $\begingroup$ That's the notation for centralizer of N in G. $\endgroup$
    – Marc Palm
    Commented Sep 3, 2013 at 5:06
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Baer in Representations of groups as quotient groups. I. Trans. Amer. Math. Soc. 58, (1945) defines the notion of commutator quotient (and he says that he takes it from Zassenhaus): if S,T are subsets of a group G he defines:

$$ S \div T = \lbrace g \in G | [T, g] \subseteq S \rbrace $$

With this notation you have

$$ Z\left( \frac{G}{N} \right) = \frac{N \div G}{N}$$

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