Characteristic Subgroup is Normal

In summary, a subgroup N of a group G is called characteristic if it remains unchanged under all automorphisms of G. If N is characteristic, it can be shown that N is a normal subgroup of G, meaning it is fixed under conjugation by any element. This can be demonstrated by considering the automorphisms of G and their corresponding elements.
  • #1
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Homework Statement
Recall that a subgroup N of a group G is called characteristic if f(N) = N for all automorphisms f of G. If N is a characteristic subgroup of G, show that N is a normal subgroup of G.

The attempt at a solution
I must show that if g is in G, then gN = Ng. Let n be in N. Since N is characteristic, there are automorphisms f and f' of G and elements a and a' of G such that n = f(a) = f'(a'). There are also elements b and b' in G such that g = f(b) = f'(b') so that gn = f(b)f(a) = f(ba) = f'(b')f(a) = f(b)f'(a') = f'(b')f'(a') = f'(b'a'). This is all I can think of and I don't see how this allows me to prove that N is normal. Am I missing something?
 
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  • #2
A subgroup is normal if and only if it is fixed (set wise) under conjugation by any element. That's precisely what gN=Ng (i.e gNg^-1 = N) says
 
  • #3
Wouldn't it be nice if f_g:G-->G: h-->ghg^-1 were an automorphism of G for each g in G?
 
  • #4
Thanks for the tip quasar987. I understand now.
 

What is a characteristic subgroup?

A characteristic subgroup is a subgroup of a group that is invariant under all automorphisms of the larger group. In other words, any automorphism of the group will map the characteristic subgroup to itself.

How is a characteristic subgroup different from a normal subgroup?

A normal subgroup is a subgroup that is invariant under conjugation by all elements of the larger group. A characteristic subgroup is a special type of normal subgroup in which the invariance is guaranteed by all automorphisms of the group. This means that a characteristic subgroup is always a normal subgroup, but the converse is not necessarily true.

Can a subgroup be both characteristic and normal?

Yes, a subgroup can be both characteristic and normal. In fact, every characteristic subgroup is also a normal subgroup, but not every normal subgroup is characteristic.

How do characteristic subgroups relate to group homomorphisms?

Characteristic subgroups are closely related to group homomorphisms. In fact, if two groups have the same characteristic subgroups, then there exists an isomorphism between the two groups. This means that characteristic subgroups are useful in studying the structure of groups and finding connections between different groups.

What is the significance of characteristic subgroups in group theory?

Characteristic subgroups play an important role in the study of group theory. They provide a useful tool for understanding the structure of groups and identifying connections between different groups. Additionally, characteristic subgroups have applications in other areas of mathematics, such as algebraic geometry and cryptography.

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