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A group that can be written as $\frac{G}{Z(G)}$ for some group $G$ is called capable. How can I find all capable groups $G$ of order 32 with $|Cent(G)|=10$, where $Cent(G)$ is the set of all centralizers of elements in the group $G$? Is it possible to find it with GAP?

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  • $\begingroup$ The numbers of the groups $G$ of order $32$ in the small groups library with $|{\rm Cent}(G)| = 10$ are $ [ 6, 7, 8, 18, 19, 20, 43, 44 ]$. You can easily do that in GAP. But you can't test for capability in any naive way, so you will need to use some of your knowledge of the properties of capable groups to get any further. $\endgroup$ – Derek Holt Sep 2 '15 at 9:28
  • $\begingroup$ I know that the SmallGroup(32,18) is $D_{32}$, that is a capable group and the SmallGroup(32,20) is $Q_{32}$, that is not a capable group. Also by comment of ULT, SmallGroup(32,6) is a capable group. $\endgroup$ – M. R. Sep 2 '15 at 9:47
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In fact you can do this directly in $\mathsf{GAP}$. There is a function $\mathtt{Epicentre}$, and a group is capable if and only if its epicentre is trivial. It turns out that only small groups number $6$ and $18$ are capable from the candidates above.

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  • $\begingroup$ Thank you. I found another function "IsCentralFactor(G)" that say that if $G$ is capable or not. $\endgroup$ – M. R. Sep 2 '15 at 11:32
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A quick search in the small groups database shows that there are only 8 groups of order 32 having 10 centralizers: the groups indexed by 6, 7, 8, 18, 19, 20, 43, 44 (see here for more information). These are 2-groups of nilpotency class 3 respectively 4. There is a necessary condition for such groups to be capable as mentioned in this excellent post but this condition is never violated here, so we cannot exclude any from this list.

I don't know about further conditions but a brute force search in the small groups database shows that at least the groups indexed by 8 and 18 are capable:

SmallGroup(32,6) is the central quotient of SmallGroup(64,32)

SmallGroup(32,18) is the central quotient of SmallGroup(64,52)

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  • $\begingroup$ I mean by $|Cent(G)|$ the number of centralizers in the group $G$. $\endgroup$ – M. R. Sep 2 '15 at 7:56
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    $\begingroup$ Even that is not remotely clear. Centralizers of what? You should put a proper formal definition in the question. $\endgroup$ – Derek Holt Sep 2 '15 at 8:24
  • $\begingroup$ I think $Cent(G) = \lbrace C_G(g) | g \in G \rbrace$ is what is meant. Correct? $\endgroup$ – Uname Sep 2 '15 at 8:26
  • $\begingroup$ Yes it is. $Cent(G)$ is the set of all centralizers in the group $G$. $\endgroup$ – M. R. Sep 2 '15 at 8:56
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    $\begingroup$ You should put the definition in the question. $\endgroup$ – Derek Holt Sep 2 '15 at 9:04

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