Finite P P P -groups Whose All Abelian Subgroups Are 2 2 2 -generated

Introduction

Finite $p$-groups are a fundamental area of study in group theory, with applications in various fields such as number theory, algebraic geometry, and computer science. A $p$-group is a group whose order is a power of a prime number $p$. In this article, we will focus on finite $p$-groups whose all abelian subgroups are $2$-generated, and explore the properties and characteristics of such groups.

Background and Motivation

It is well known that a finite $p$-group in which all abelian subgroups are cyclic (equivalently, in which $\Omega_1(G)$ has order $p$), is either cyclic or a generalized quaternion group. This result has been extensively studied and generalized in various ways. However, the question of whether there exist finite $p$-groups whose all abelian subgroups are $2$-generated remains an open problem.

Definition and Notation

Before we proceed, let us define some notation and terminology. Let $G$ be a finite $p$-group. We denote by $\Omega_1(G)$ the subgroup generated by all elements of $G$ of order $p$. We say that a subgroup $H$ of $G$ is $2$-generated if there exist two elements $x, y \in H$ such that $H = \langle x, y \rangle$. We denote by $Z(G)$ the center of $G$, and by $C_G(x)$ the centralizer of $x$ in $G$.

Properties of Finite $p$-groups

Finite $p$-groups have several important properties that make them amenable to study. One of the most fundamental properties is the fact that every finite $p$-group has a non-trivial center. This is a consequence of the fact that the center of a $p$-group is a characteristic subgroup, and therefore is invariant under all automorphisms of the group.

Another important property of finite $p$-groups is the fact that they have a non-trivial Frattini subgroup. The Frattini subgroup of a group $G$ is the intersection of all maximal subgroups of $G$. In the case of a finite $p$-group, the Frattini subgroup is a characteristic subgroup, and therefore is invariant under all automorphisms of the group.

Abelian Subgroups of Finite $p$-groups

Abelian subgroups of finite $p$-groups are a crucial area of study in the theory of $p$-groups. As we mentioned earlier, it is well known that a finite $p$-group in which all abelian subgroups are cyclic is either cyclic or a generalized quaternion group. However, the question of whether there exist finite $p$-groups whose all abelian subgroups are $2$-generated remains an open problem.

Characterization of Finite $p$-groups with $2$-generated Abelian Subgroups

In this section, we will provide a characterization of finite $p$-groups whose all abelian subgroups are $2$-generated. We will show that such groups are precisely those in which every nonabelian subgroup has order $p^3$.

Theorem 1

Let $G$ be a finite $p$-group. Suppose that every abelian subgroup of $G$ is $2$-generated. Then every non-abelian subgroup of $G$ has order $p^3$.

Proof

Let $H$ be a non-abelian subgroup of $G$. Suppose that $|H| > p^3$. Then $H$ has a subgroup $K$ of order $p^2$ such that $K$ is normal in $H$. Since $K$ is normal in $H$, it is also normal in $G$. Therefore, $K$ is an abelian subgroup of $G$. However, since $K$ is not $2$-generated, this contradicts the assumption that every abelian subgroup of $G$ is $2$-generated. Therefore, every non-abelian subgroup of $G$ has order $p^3$.

Corollary 1

Let $G$ be a finite $p$-group. Suppose that every abelian subgroup of $G$ is $2$-generated. Then $G$ is a $p$-group of maximal class.

Proof

Let $G$ be a finite $p$-group such that every abelian subgroup of $G$ is $2$-generated. Then every non-abelian subgroup of $G$ has order $p^3$ by Theorem 1. Therefore, $G$ is a $p$-group of maximal class.

Open Problems

Despite the progress made in this article, there are still many open problems in the theory of finite $p$-groups whose all abelian subgroups are $2$-generated. Some of the most pressing open problems include:

• Problem 1: Characterize the structure of finite $p$-groups whose all abelian subgroups are $2$-generated.
• Problem 2: Determine the number of non-isomorphic finite $p$-groups whose all abelian subgroups are $2$-generated.
• Problem 3: Investigate the properties of finite $p$-groups whose all abelian subgroups are $2$-generated in relation to other areas of group theory, such as the theory of nilpotent groups and the theory of solvable groups.

Conclusion

Q: What is a finite $p$-group?

A: A finite $p$-group is a group whose order is a power of a prime number $p$. In other words, it is a group whose order is of the form $p^n$, where $n$ is a positive integer.

Q: What is the significance of $2$-generated abelian subgroups in finite $p$-groups?

A: In a finite $p$-group, an abelian subgroup is $2$-generated if there exist two elements in the subgroup such that the subgroup is generated by these two elements. The significance of $2$-generated abelian subgroups lies in the fact that they provide a way to understand the structure of the group.

Q: What is the relationship between $2$-generated abelian subgroups and the center of a finite $p$-group?

A: The center of a group is the set of elements that commute with every other element in the group. In a finite $p$-group, the center is a characteristic subgroup, and therefore is invariant under all automorphisms of the group. The relationship between $2$-generated abelian subgroups and the center of a finite $p$-group is that the center is a $2$-generated abelian subgroup.

Q: What is the Frattini subgroup of a finite $p$-group?

A: The Frattini subgroup of a group is the intersection of all maximal subgroups of the group. In a finite $p$-group, the Frattini subgroup is a characteristic subgroup, and therefore is invariant under all automorphisms of the group.

Q: What is the relationship between $2$-generated abelian subgroups and the Frattini subgroup of a finite $p$-group?

A: The relationship between $2$-generated abelian subgroups and the Frattini subgroup of a finite $p$-group is that the Frattini subgroup is a $2$-generated abelian subgroup.

Q: What is the significance of Theorem 1 in the context of finite $p$-groups?

A: Theorem 1 states that every non-abelian subgroup of a finite $p$-group has order $p^3$ if every abelian subgroup of the group is $2$-generated. This theorem is significant because it provides a characterization of finite $p$-groups with $2$-generated abelian subgroups.

Q: What is the relationship between finite $p$-groups with $2$-generated abelian subgroups and $p$-groups of maximal class?

A: The relationship between finite $p$-groups with $2$-generated abelian subgroups and $p$-groups of maximal class is that every finite $p$-group with $2$-generated abelian subgroups is a $p$-group of maximal class.

Q: What are some open problems in the context of finite $p$-groups with $2$-generated abelian subgroups?

A: Some open problems in the context of finite $p$-groups with $2$-generated abelian subgroups include:

• Problem 1: Characterize the structure of finite $p$-groups with $2$-generated abelian subgroups.
• Problem 2: Determine the number of non-isomorphic finite $p$-groups with $2$-generated abelian subgroups.
• Problem 3: Investigate the properties of finite $p$-groups with $2$-generated abelian subgroups in relation to other areas of group theory, such as the theory of nilpotent groups and the theory of solvable groups.

Conclusion

In this Q&A article, we have explored some of the key concepts and results related to finite $p$-groups with $2$-generated abelian subgroups. We have also identified some open problems in this area of study. Further research is needed to fully understand the properties and characteristics of finite $p$-groups with $2$-generated abelian subgroups.