My question is about the shaded area in this image. 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 this paper.

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