rapple said:WTS For any g in D-n, g(r^kf)g^-1 Not In <F>
A subgroup of Dn is a subset of the group Dn that is also a group under the same operation. It contains the identity element, and every element in the subgroup has an inverse within the subgroup.
A subgroup is normal if it is invariant under conjugation by elements of the larger group. This means that for any element g in the larger group and any element h in the subgroup, the element ghg-1 is also in the subgroup.
To prove that a subgroup is not normal, you can show that there exists an element in the larger group that, when conjugated with an element in the subgroup, does not result in an element within the subgroup. This is a counterexample that disproves the subgroup's normality.
The generating element f is used to construct the subgroup in question. It is often chosen as a convenient and easily identifiable element that can generate the entire subgroup. This allows for a simpler and more straightforward proof of the subgroup's normality or lack thereof.
No, a subgroup of Dn can only be normal for certain values of n. For example, it can be shown that all subgroups of D3 are normal, but this is not the case for all values of n. For instance, subgroups of D4 are not always normal.