Quick question about Automorphisms

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In summary: Your Name]In summary, an automorphism p on a group G will preserve the intersection of two subgroups, meaning that p(H ∩ K) = p(H) ∩ p(K). This can be shown by proving that p(H ∩ K) ⊆ p(H) ∩ p(K) and p(H) ∩ p(K) ⊆ p(H ∩ K). By using the bijective property of p and its inverse, we can show that the elements in p(H) ∩ p(K) are also in p(H ∩ K), and vice versa. This proves that the two sets are equal.
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1MileCrash
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If p is an automorphism on G, and H and K are subgroups of G, does p(H intersect K) = p(H) intersect p(K)?

If so, how can I show this?

EDIT: nevermind
 
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, I figured it out!

Hello,

Yes, it is true that p(H ∩ K) = p(H) ∩ p(K) if p is an automorphism on G. This is because an automorphism is a bijective map that preserves the group structure, so it will preserve the intersection of two subgroups.

To show this, we can first prove that p(H ∩ K) ⊆ p(H) ∩ p(K). Let x be an element of p(H ∩ K). Then, x = p(y) for some y ∈ H ∩ K. Since y ∈ H ∩ K, it is in both H and K. Therefore, p(y) ∈ p(H) and p(y) ∈ p(K), which means that p(y) ∈ p(H) ∩ p(K). This shows that p(H ∩ K) ⊆ p(H) ∩ p(K).

Next, we need to prove that p(H) ∩ p(K) ⊆ p(H ∩ K). Let x be an element of p(H) ∩ p(K). Then, x = p(y) for some y ∈ H and x = p(z) for some z ∈ K. Since p is a bijective map, p^-1 exists and is also a bijective map. Therefore, p^-1(x) = y and p^-1(x) = z. Since y ∈ H and z ∈ K, p^-1(x) = y ∈ H ∩ K. This means that x = p(p^-1(x)) ∈ p(H ∩ K), which shows that p(H) ∩ p(K) ⊆ p(H ∩ K).

Combining these two proofs, we can conclude that p(H ∩ K) = p(H) ∩ p(K). I hope this helps!

 

What are Automorphisms?

Automorphisms are mathematical transformations that preserve the structure and properties of a mathematical object. They are essentially "symmetries" of the object.

What is the importance of Automorphisms?

Automorphisms are important in mathematics as they help in understanding the underlying structure and properties of a mathematical object. They also have a wide range of applications in various fields, such as group theory, graph theory, and algebraic geometry.

How do you find Automorphisms?

The process of finding automorphisms depends on the specific mathematical object in question. In general, it involves analyzing the structure and properties of the object and determining which transformations preserve those properties.

Can Automorphisms exist for any mathematical object?

No, not all mathematical objects have automorphisms. For example, a line segment does not have any non-trivial automorphisms, while a square has four. The existence of automorphisms depends on the structure and properties of the object.

What is the difference between an Automorphism and an Isomorphism?

An automorphism is a transformation that preserves the structure and properties of a mathematical object, while an isomorphism is a type of automorphism that is also bijective (one-to-one and onto). In other words, all isomorphisms are automorphisms, but not all automorphisms are isomorphisms.

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