]. Show that [itex]a^m\in H[/itex] for all a[itex]\in[/itex] G.An elementary cyclic normal group is a type of group in abstract algebra that is both cyclic (meaning it can be generated by a single element) and normal (meaning its subgroups are also normal). It is a fundamental concept in group theory and can be found in various mathematical fields, such as number theory and geometry.
An elementary cyclic normal group has the following properties:
An elementary cyclic normal group is a special case of a cyclic group. While all elementary cyclic normal groups are cyclic, not all cyclic groups are elementary cyclic normal groups. The key difference is that an elementary cyclic normal group must also be normal, while a cyclic group does not necessarily have this property.
Elementary cyclic normal groups are important in abstract algebra and other mathematical fields because they provide a simpler and more structured understanding of groups. They also have many useful properties and can be used to prove theorems and solve problems in various areas of mathematics.
No, an elementary cyclic normal group cannot have non-abelian subgroups. This is because all of its subgroups are normal, and a non-abelian subgroup would not be normal in an abelian group. Additionally, an elementary cyclic normal group is itself abelian, so it cannot contain any non-abelian subgroups.