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My question is about the shaded area in this image.

enter image description here

Does the symbol $L=\bigcup_{g \in G} T^{g}$ means that $L$ is a union of sets or $L=\langle T^{g}, g\in G \rangle$? If it means the first one, then how did the authors prove that $L$ equals to that union? If it means the second one, then how did the author conclude that all the elements of order p or 4 (if $p=2$) of $L$ are contained in $K$? I like to add that $L/K$ is an abelian p-group of exponent $p$.

My question comes from the the paper BaoJun Li, ZhiRang Zhang: The influence of s-conditionally permutable subgroups on finite groups, Sci. China Ser. A-Math. (2009) 52: 301; DOI: 10.1007/s11425-009-0028-4

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  • $\begingroup$ The symbol has the same meaning as in other areas of math: I.e. $L$ (as a set) equals the union of the sets $T^g$. Group theory doesn't redefine this notation. $\endgroup$
    – Max Horn
    Commented Aug 26, 2012 at 21:44
  • $\begingroup$ @Max: I think some people used the second meaning for that symbole. This link is an example (ams.org/journals/bull/1934-40-12/S0002-9904-1934-05982-2/… ). Please see the second paragraph. $\endgroup$
    – user23954
    Commented Aug 26, 2012 at 22:51
  • $\begingroup$ I mean the third paragraph. $\endgroup$
    – user23954
    Commented Aug 26, 2012 at 22:54
  • $\begingroup$ @unknown(google) could you please write down the question? I could not open the file as it is diverted to a site..... Thanks $\endgroup$
    – Alireza Abdollahi
    Commented Aug 28, 2012 at 15:05
  • $\begingroup$ @Alireza Abdollahi: I sent the image to your email. $\endgroup$
    – user23954
    Commented Aug 28, 2012 at 15:57

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