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Could a finitely generated group contain a subgroup which is infinitely generated? Why?
A finitely generated group is a mathematical object that is made up of a finite number of elements and operations. These elements and operations follow certain rules and properties, which allow them to be combined and manipulated in various ways.
Finitely generated groups are different from other groups in that they have a finite number of elements, whereas other groups may have infinitely many elements. Additionally, the elements of a finitely generated group can be expressed as a combination of a finite set of generators, while other groups may not have such a simple form.
Some common examples of finitely generated groups include cyclic groups, which are generated by a single element, and dihedral groups, which are generated by two elements. Other examples include finite abelian groups and certain types of free groups.
Finitely generated groups are used in mathematics to study and understand various structures and properties. They can be applied to fields such as algebra, geometry, and topology, and have real-world applications in areas such as cryptography and computer science.
No, not all groups are finitely generated. Some groups, such as the group of real numbers under addition, have infinitely many elements and cannot be expressed as a combination of a finite set of generators. However, many important and interesting groups are finitely generated and are studied extensively in mathematics.