A counter example is a specific instance or case that contradicts a general statement or rule. It is used to disprove a hypothesis or claim.
An Abelian subgroup is a subgroup of a group where the elements commute with each other. This means that the order in which the elements are multiplied does not affect the result.
A normal subgroup is a subgroup that is invariant under conjugation by elements of the larger group. In other words, it is a subgroup that remains unchanged when the elements of the larger group are multiplied or inverted.
No, an Abelian subgroup cannot be a normal subgroup. This is because an Abelian subgroup is defined by the commutativity of its elements, while a normal subgroup is defined by the invariance of its elements under conjugation. These two definitions are contradictory.
Yes, consider the group of integers under addition. The subgroup of even numbers is Abelian, as any two even numbers added together will always result in an even number. However, this subgroup is not normal since conjugation by an odd number will result in an odd number, which is not in the subgroup.