How does the author determine the elements of order p or 4 in the group?

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The discussion centers on the determination of elements of order p or 4 in the group L, specifically how these elements are contained in the normal subgroup K. The author demonstrates that for any element x in L, there exists a corresponding element y in T such that x can be expressed as x = gyg^{-1}, where g is an element of G. Since y is contained in K and retains the same order as x, it follows that x must also be in K. The notation L = ∪_{g ∈ G} T^g indicates that L is defined as the union of the conjugates of T under G.

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My question is about the shaded area in the attachment?
How did the author get that all the elements of order p or 4 of L are contained in K? I mentioned the abstract but I do not think there is a need for that.

Help?
 

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Take x\in L. Then there is a g in G such that x\in T^g and thus there is a y in T such that x=gyg^{-1}. If x has order p or order 4, then so does y. But by the previous sentence, y is contained in K. Since K is normal, we have that also x is in K.
 
The symbol L=\bigcup_{g \in G} T^{g}, does it mean the union of sets or L=<T^{g},g \in G>and, if it the union of sets, then how did he gets that L equals to that union?
 

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