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Currently I am studying the definition of Socle of a group. Also, I came to know that Socle of a finite nilpotent group $G$ is the product of elementary abelian $p$-groups for the collection of primes dividing the order of $G$. I know that Hamiltonian groups are nilpotent of class $2$. Now, I am wondering what can be the Socle of a Hamiltonian group. More precisely, I have the following question.

Let $H$ be an infinite Hamiltonian group, that is, $H$ is an infinite non-abelian group all of whose subgroups are normal. What is the Socle of $H$, $\mathrm{Soc}(H)$?

Debarati
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  • Welcome to MSE! Please share what you have tried – beingmathematician Jun 23 '22 at 19:01
  • I know that infinite non-abelian Hamiltonian group H is the direct sum of a quaternion group Q8, an elementary abelian 2-group B and a torsion abelian group with all elements of odd order D. From here, I am trying to show that Socle is the direct product of cyclic group of order 2, B and some subgroup of D. – Debarati Jun 23 '22 at 19:11
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    This does have context, though, @beingmathematician; see here. – Shaun Jun 23 '22 at 19:24
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    In general ${\rm Soc}(G \times H) = {\rm Soc}(H) \times {\rm Soc}(H)$. – Derek Holt Jun 23 '22 at 19:26
  • I didn't know that held in general, @DerekHolt. Where might I find a proof? I couldn't find it anywhere . . . – Shaun Jun 23 '22 at 19:37
  • @DerekHolt Can you please provide references so that I can see the proof of the in general result that youposted? – Debarati Jun 23 '22 at 20:06
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    I searched for "socle of direct product of groups" and found that I had already answered this question here! That question was about finite groups, but the proof does assume finiteness. – Derek Holt Jun 23 '22 at 20:54
  • Thank you, @DerekHolt! – Shaun Jun 23 '22 at 21:00
  • @DerekHolt Can we apply the result then for this problem regarding Hamiltonian group? – Debarati Jun 23 '22 at 22:12
  • Yes we can, we can apply it to the three direct factors of $H$ that you mentioned and derive the result you want. ${\rm Soc}(Q_8) = C_2$, ${\rm Soc}(B)=B$, and ${\rm Soc}(D)$ is generated by the elements of prime order in $D$. – Derek Holt Jun 24 '22 at 08:43
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    Can you provide me some references where I can find the result, namely, $Soc(G \times H) = Soc(G) \times Soc(H)$? I understand the proof of yours in math stack exchange. But I am surprised that I could not find its references in any of the textbooks. – Debarati Jun 28 '22 at 19:27

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