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

In summary, the author explains that in order to show that all elements of order p or 4 of L are contained in K, they take an element x in L and find a corresponding element y in T that is also in K. This is possible because of the normality of K. The symbol L=\bigcup_{g \in G} T^{g} represents the union of sets and it is used to show that L is equal to the set of all T^{g} for every g in G.
  • #1
moont14263
40
0
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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  • #2
Take [itex]x\in L[/itex]. Then there is a g in G such that [itex]x\in T^g[/itex] and thus there is a y in T such that [itex]x=gyg^{-1}[/itex]. 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.
 
  • #3
The symbol [itex]L=\bigcup_{g \in G} T^{g}[/itex], does it mean the union of sets or [itex]L=<T^{g},g \in G>[/itex]and, if it the union of sets, then how did he gets that [itex]L[/itex] equals to that union?
 

What is a chief factor of a finite group?

A chief factor of a finite group is a normal subgroup that is maximal with respect to being nontrivial and not equal to the entire group. Essentially, it is a subgroup that cannot be further divided into smaller normal subgroups.

How do you determine the chief factors of a finite group?

To determine the chief factors of a finite group, you can use the chief series, which is a sequence of normal subgroups that eventually leads to the trivial group. The chief factors are the nontrivial normal subgroups in this series.

Why are chief factors important in the study of finite groups?

Chief factors are important because they provide a way to break down a finite group into its fundamental building blocks. By understanding the chief factors of a group, we can gain insight into the structure and properties of the group as a whole.

Can a finite group have more than one chief factor?

Yes, a finite group can have multiple chief factors. In fact, it is common for a finite group to have several chief factors, each of which is a distinct normal subgroup.

Are chief factors unique to finite groups?

No, chief factors can also be defined for infinite groups. However, in the study of finite groups, chief factors are particularly useful because they provide a way to classify and understand the structure of these groups in a finite setting.

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