How Many Subgroups of Order Three Does a Group Have?

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SUMMARY

The group G contains exactly eight elements of order three, which leads to the conclusion that G has four distinct subgroups of order three. Each subgroup is generated by one of the four elements, with the remaining four elements being the squares of these generators. This understanding clarifies the relationship between elements of order three and their corresponding subgroups, confirming that not every element of order three constitutes a separate subgroup.

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[SOLVED] Subgroups of Order Three

Homework Statement
Let G be a subgroup containing exactly eight elements of order three. How many subgroups of order three does G have?

The attempt at a solution
This problem was discussed in class today. The professor said that G has four subgroups of order three. I didn't follow his explanation very well so I didn't understand why. Since there are eight elements of order three, wouldn't each of these elements constitute a subgroup of order three so G has at least eight subgroups of order three?
 
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Suppose [tex]a \in G[/tex] is order 3 then [tex]a^2[/tex] is also order 3. They belong to the same subgroup. That means that only 4 of the 8 are generators, and the other 4 are their squares.
 
I see. I overlooked that fact. Thanks a lot.
 

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