Are There Groups G = ⟨ A , B ⟩ G=\langle A, B\rangle G = ⟨ A , B ⟩ Such That O ( G ) ≤ O ( A ) O ( B ) O(G) \leq O(a)o(b) O ( G ) ≤ O ( A ) O ( B ) ?

Are there groups $G=\langle a, b\rangle$ such that $o(G) \leq o(a)o(b)$?

In the realm of group theory, the study of finitely generated groups has led to numerous interesting questions and counterexamples. One such question revolves around the order of a group generated by two elements, $a$ and $b$. Specifically, we aim to investigate whether there exist groups $G=\langle a, b\rangle$ such that the order of $G$ is less than or equal to the product of the orders of $a$ and $b$, i.e., $o(G) \leq o(a)o(b)$. This inquiry is closely related to the concept of the least common multiple (LCM) of the orders of $a$ and $b$, denoted as $\text{lcm}\left(o(a), o(b)\right)$.

The study of group theory has a rich history, with contributions from mathematicians such as Évariste Galois, William Burnside, and Emmy Noether. The concept of a finitely generated group, where a group is generated by a finite set of elements, has been a central theme in group theory. In this context, the order of a group, denoted as $o(G)$, is the number of elements in the group. The study of the order of a group has led to various results and counterexamples, including the famous result that the order of a group is not necessarily equal to the product of the orders of its generators.

One of the earliest counterexamples to the question at hand is the dihedral group $D_6$, which is a group of order 12 generated by two elements, $a$ and $b$, of orders 2 and 3, respectively. This counterexample shows that the order of a group generated by two elements is not necessarily equal to the product of the orders of its generators. In fact, $D_6$ has a more complex structure, with a presentation given by $\langle a, b \mid a^2 = b^3 = (ab)^2 = 1 \rangle$. This presentation highlights the non-abelian nature of $D_6$, which is a key feature of many groups.

The concept of the least common multiple (LCM) of two numbers is a fundamental idea in number theory. Given two positive integers $m$ and $n$, the LCM of $m$ and $n$, denoted as $\text{lcm}(m, n)$, is the smallest positive integer that is divisible by both $m$ and $n$. In the context of group theory, the LCM of the orders of two elements, $a$ and $b$, is denoted as $\text{lcm}\left(o(a), o(b)\right)$. This concept is closely related to the order of a group generated by two elements, as we will see in the following section.

Let $G$ be a group generated by two elements, $a$ and $b$, i.e., $G = \langle a, b \rangle$. The order of $G$, denoted as $o(G)$, is the number of in the group. In general, the order of $G$ is not necessarily equal to the product of the orders of $a$ and $b$, i.e., $o(G) \leq o(a)o(b)$. However, there are some cases where the order of $G$ is equal to the product of the orders of $a$ and $b$. For example, if $a$ and $b$ are two elements of a group $G$ such that $ab = ba$, then the order of $G$ is equal to the product of the orders of $a$ and $b$.

The relationship between the order of a group and the LCM of the orders of its generators is a fundamental question in group theory. In general, the order of a group generated by two elements is not necessarily equal to the LCM of the orders of its generators. However, there are some cases where the order of a group is equal to the LCM of the orders of its generators. For example, if $G$ is a group generated by two elements, $a$ and $b$, such that $o(G) = \text{lcm}\left(o(a), o(b)\right)$, then $G$ is said to be a group with a "nice" order.

The question of whether there exist groups $G=\langle a, b\rangle$ such that $o(G) \leq o(a)o(b)$ remains an open question in group theory. While there are some counterexamples, such as the dihedral group $D_6$, there are also some cases where the order of a group is equal to the product of the orders of its generators. Further research is needed to fully understand the relationship between the order of a group and the LCM of the orders of its generators.

In conclusion, the question of whether there exist groups $G=\langle a, b\rangle$ such that $o(G) \leq o(a)o(b)$ is a fundamental question in group theory. While there are some counterexamples, such as the dihedral group $D_6$, there are also some cases where the order of a group is equal to the product of the orders of its generators. Further research is needed to fully understand the relationship between the order of a group and the LCM of the orders of its generators.

• Burnside, W. (1911). Theory of Groups of Finite Order. Cambridge University Press.
• Hall, P. (1959). The Theory of Groups. Macmillan.
• Rotman, J. J. (1995). An Introduction to the Theory of Groups. Springer-Verlag.

The following is a list of some of the key concepts and results mentioned in this article:

• Finitely generated group: A group that can be generated by a finite set of elements.
• Order of a group: The number of elements in a group.
• Least common multiple (LCM): The smallest positive integer that is divisible by two or more positive integers.
• Dihedral group: A group of order 12 generated by two elements, $a$ and $b$, of orders 2 and 3, respectively* Presentation of a group: A way of describing a group using generators and relations.
  Q&A: Are there groups $G=\langle a, b\rangle$ such that $o(G) \leq o(a)o(b)$?

Q: What is the significance of the question "Are there groups $G=\langle a, b\rangle$ such that $o(G) \leq o(a)o(b)$?"

A: The question is significant because it deals with the relationship between the order of a group generated by two elements and the product of the orders of its generators. Understanding this relationship can provide insights into the structure of groups and their properties.

Q: What is the least common multiple (LCM) of the orders of two elements?

A: The LCM of the orders of two elements, $a$ and $b$, is the smallest positive integer that is divisible by both $o(a)$ and $o(b)$. It is denoted as $\text{lcm}\left(o(a), o(b)\right)$.

Q: Can you provide an example of a group where the order of the group is equal to the product of the orders of its generators?

A: Yes, consider the group $G = \langle a, b \mid a^2 = b^3 = (ab)^2 = 1 \rangle$. In this group, the order of $a$ is 2 and the order of $b$ is 3. The order of the group $G$ is 6, which is equal to the product of the orders of $a$ and $b$.

Q: What is the relationship between the order of a group and the LCM of the orders of its generators?

A: In general, the order of a group generated by two elements is not necessarily equal to the LCM of the orders of its generators. However, there are some cases where the order of a group is equal to the LCM of the orders of its generators.

Q: Can you provide a counterexample to the question "Are there groups $G=\langle a, b\rangle$ such that $o(G) \leq o(a)o(b)$?"

A: Yes, consider the dihedral group $D_6$, which is a group of order 12 generated by two elements, $a$ and $b$, of orders 2 and 3, respectively. In this group, the order of $D_6$ is 12, which is greater than the product of the orders of $a$ and $b$.

Q: What is the significance of the dihedral group $D_6$ in the context of this question?

A: The dihedral group $D_6$ is a counterexample to the question "Are there groups $G=\langle a, b\rangle$ such that $o(G) \leq o(a)o(b)$?" It shows that the order of a group generated by two elements is not necessarily equal to the product of the orders of its generators.

Q: What are some open questions and future directions in this area of research?

A: Some open questions and future directions in this area of research include:

• Investigating the relationship between the order of a group and the LCM of the orders of its generators in more general cases.
• Studying the properties of groups with a "nice" order, i.e., groups where the order of the group is equal to the LCM of the orders of its generators.
• Exploring implications of the dihedral group $D_6$ and other counterexamples on our understanding of group theory.

Q: What are some key concepts and results mentioned in this article?

A: Some key concepts and results mentioned in this article include:

• Finitely generated group: A group that can be generated by a finite set of elements.
• Order of a group: The number of elements in a group.
• Least common multiple (LCM): The smallest positive integer that is divisible by two or more positive integers.
• Dihedral group: A group of order 12 generated by two elements, $a$ and $b$, of orders 2 and 3, respectively.
• Presentation of a group: A way of describing a group using generators and relations.