Using class equation to show that intersection is nontrivial

In summary, the class equation is used to relate the number of elements in a group to the number of elements in its distinct conjugacy classes. It can be used to show that the intersection of two subgroups is nontrivial by proving that the number of elements in the intersection is greater than 1. A nontrivial intersection refers to the situation where two subgroups of a group have at least one element in common. The class equation can also be used to show that the intersection is trivial by proving that the number of elements in the intersection is equal to 1. There are no specific conditions for using the class equation to show nontrivial intersection. The class equation has practical applications in various areas of mathematics, including number theory,
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Mr Davis 97
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Homework Statement


Let ##G## be a finite p-group and ##Z(G)## its center. If ##N \not = \{e\}## is a normal subgroup of ##G##, prove that ##N\cap Z(G) \not = \{e\}##.

Homework Equations

The Attempt at a Solution


Since ##N## is a normal subgroup we can let ##G## act on ##N## by conjugation. In a manner similar to the case when ##G## acts on itself, we can construct the following class equation. Let ##n_1,\dots,n_r## be representatives of the orbits of this action not contained in ##N\cap Z(G)##. Then $$|G| = |N\cap Z(G)| + \sum_{i=1}^{r}[G : \operatorname{Stab}_G(n_i)].$$ Since some prime ##p## divides ##|G|## and ##[G : \operatorname{Stab}_G(n_i)]## for all ##i\in [1,r]##, it follows that ##p## divides ##|N\cap Z(G)|##. Hence ##N \cap Z(G) \not = \{e\}##. QED

My main question is have I explained in enough depth how I obtain the class equation that I got? Do I need to show in a rigorous way that $$|G| = |N\cap Z(G)| + \sum_{i=1}^{r}[G : \operatorname{Stab}_G(n_i)],$$ or is what I have written sufficient?
 
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I think it is sufficient.

The stabilizer formula is usually formulated for a single element. So if you like, you can show how the single element version ##|G|=|G.x|\cdot |C_G(x)|=|G:C_G(x)|\cdot |C_G(x)|## transforms into a version for entire ##N##. You used that different orbits are either disjoint or equal to obtain the sum, and that all fix points of ##N## are central. But your version is o.k.
 
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FAQ: Using class equation to show that intersection is nontrivial

1. How is the class equation used to show that intersection is nontrivial?

The class equation is a mathematical formula that relates the number of elements in a group to the number of elements in its distinct conjugacy classes. If the intersection of two subgroups is nontrivial, it means that the two subgroups have at least one common element. By using the class equation, we can show that the number of elements in the intersection is greater than 1, thus proving that the intersection is nontrivial.

2. What is a nontrivial intersection?

A nontrivial intersection refers to the situation where two subgroups of a group have at least one element in common. In other words, the intersection is not empty and contains elements other than the identity element. It is an important concept in group theory as it helps us understand the relationship between different subgroups of a group.

3. Can the class equation be used to show that the intersection of two subgroups is trivial?

Yes, the class equation can also be used to show that the intersection of two subgroups is trivial. In this case, we would show that the number of elements in the intersection is equal to 1, meaning that the only common element between the two subgroups is the identity element. This proves that the intersection is trivial.

4. Are there any specific conditions for using the class equation to show nontrivial intersection?

There are no specific conditions for using the class equation to show nontrivial intersection. As long as the group is finite and the subgroups in question are proper subgroups, the class equation can be used to show whether the intersection is nontrivial or trivial.

5. What are some practical applications of using the class equation to show nontrivial intersection?

The class equation and the concept of nontrivial intersection have applications in various areas of mathematics, including number theory, geometry, and algebraic topology. In particular, they are useful in understanding the structure and properties of finite groups, as well as in solving problems related to symmetry and group actions.

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