micromass said:The statement you write is true. How would you find h?? Hint: rewrite the equation [itex]n=a^{-1}ha[/itex] to find an expression for h.
AdrianZ said:well, then is this argument valid?
first I should show that N is a subgroup of G, it means I should prove that for any a and b in N, ab-1 is also in N.
[itex]\forall p,b \in N: \forall a \in G, \exists h_1,h_2 \in H : p=a^{-1}h_1a , b=a^{-1}h_2a[/itex]
[itex] pb^{-1}= a^{-1}h_1a(a^{-1}h_2a)^{-1}=a^{-1}h_1a(a)^{-1}h_2^{-1}(a^{-1})^{-1}=a^{-1}hh^{-1}a \implies \exists h_3 \in H, \forall a \in G: a^{-1}h_3a=pb^{-1}[/itex]
hence, [itex]pb^{-1} \in N[/itex] and N is a subgroup of G.
If that argument is valid, how can I show that N is normal?
A normal subgroup is a subgroup that is invariant under conjugation by elements of the larger group. In other words, if we take any element in the normal subgroup and conjugate it by any element in the larger group, the resulting element will still be in the normal subgroup. This is also known as being "normalizer-invariant".
To prove that a subgroup N is normal, we must show that for every element g in the larger group G, the conjugate of N by g (denoted gNg^-1) is still equal to N. This can be done by showing that every element in N commutes with every element in G, or by showing that the left and right cosets of N in G are equal.
Normal subgroups play an important role in group theory, as they are closely related to the concept of quotient groups. If N is a normal subgroup of G, then we can form the quotient group G/N, which is a new group whose elements are the cosets of N in G. Normal subgroups also allow us to classify groups into different types, such as abelian and non-abelian groups.
No, a subgroup cannot be both normal and non-normal at the same time. A subgroup is either normal or not normal in a particular group. However, a subgroup can be normal in one group and non-normal in another group.
If a subgroup N is normal in a group G, then we can use the normality to simplify certain group operations. For example, the quotient group G/N inherits many properties from G, making it easier to study. Additionally, the normality of a subgroup can provide insight into the structure of the larger group, such as determining if the group is solvable or simple.