Normalcy in a finite group refers to a subgroup that remains unchanged when conjugated by any element in the group. In other words, the subgroup is invariant under conjugation.
To prove normalcy of H in a finite group G, you must show that the subgroup H is invariant under conjugation by any element in G. This can be done by showing that for any element h in H and any element g in G, the conjugate ghg^-1 is also in H.
Proving normalcy of H in a finite group G is important because it allows us to study the structure of the group G by breaking it down into smaller, more manageable subgroups. It also helps in identifying important properties and relationships between the elements in G.
Yes, a finite group can have multiple normal subgroups. In fact, every finite group has at least two normal subgroups - the trivial subgroup {e} and the group itself G.
Some common techniques used to prove normalcy of H in a finite group G include direct proof, proof by induction, and using the properties of normal subgroups such as closure under conjugation and quotient groups.