The Condition of Commutativity

Oct 13, 2025 by windailys.com 131 views

Introduction

In group theory, a product group is a way of combining two groups to form a new group. Given two groups $K$ and $H$ , the product group $K \times H$ is defined as the set of ordered pairs $(k, h)$ , where $k \in K$ and $h \in H$ , with the operation of component-wise multiplication. In this article, we will explore the conditions under which a finite group $G$ can be written as a product group $K \times H$ , where $K$ and $H$ are groups of strictly smaller order.

The Order of a Group

The order of a group is the number of elements in the group. For a group $G$ to be written as a product group $K \times H$ , the order of $G$ must be the product of the orders of $K$ and $H$ . This is because each element of $G$ can be uniquely represented as an ordered pair $(k, h)$ , where $k \in K$ and $h \in H$ . Therefore, the order of $G$ is the product of the number of elements in $K$ and the number of elements in $H$ .

The Condition of Non-Primality

One of the most important conditions for a group $G$ to be written as a product group $K \times H$ is that the order of $G$ must not be prime. If the order of $G$ is prime, then $G$ is a cyclic group, and it cannot be written as a product group of two smaller groups. This is because a cyclic group of prime order has no proper subgroups, and therefore, it cannot be decomposed into a product of two smaller groups.

The Condition of Commutativity

Another important condition for a group $G$ to be written as a product group $K \times H$ is that the groups $K$ and $H$ must commute with each other. This means that for any elements $k \in K$ and $h \in H$ , the following equation must hold:

k \cdot h = h \cdot k

This condition is necessary because the product group $K \times H$ is defined as the set of ordered pairs $(k, h)$ , and the operation of component-wise multiplication is used to combine these pairs. If $K$ and $H$ do not commute with each other, then the product group $K \times H$ will not be well-defined.

The Condition of Direct Product

A group $G$ can be written as a product group $K \times H$ if and only if $G$ is a direct product of $K$ and $H$ . This means that $G$ must have a subgroup $K$ and a subgroup $H$ such that:

$K \cap H = \{e\}$ , where $e$ is the identity element of $G$ .
$KH = G$ , where $KH$ is the set of all products $kh$ , with $k \in K$ and $h \in H$ .

This condition is necessary because the product group $K \times H$ is defined as the set of ordered pairs $(k, h)$ , and the operation of component-wise multiplication is used to combine these pairs. If $G$ is not a direct product of $K$ and $H$ , then the product group $K \times H$ will not be well-defined.

Examples of Product Groups

There are many examples of product groups in group theory. For instance, the group of integers modulo $n$ can be written as a product group of the group of integers modulo $m$ and the group of integers modulo $k$ , where $n = mk$ . Another example is the group of $2 \times 2$ invertible matrices over the real numbers, which can be written as a product group of the group of $2 \times 2$ invertible matrices over the real numbers with determinant $1$ and the group of $2 \times 2$ invertible matrices over the real numbers with determinant $-1$ .

Conclusion

Q: What is a product group in group theory?

A: A product group is a way of combining two groups to form a new group. Given two groups $K$ and $H$ , the product group $K \times H$ is defined as the set of ordered pairs $(k, h)$ , where $k \in K$ and $h \in H$ , with the operation of component-wise multiplication.

Q: What are the conditions for a group to be written as a product group?

A: A group $G$ can be written as a product group $K \times H$ if and only if the order of $G$ is not prime, the groups $K$ and $H$ commute with each other, and $G$ is a direct product of $K$ and $H$ .

Q: What is the order of a product group?

A: The order of a product group $K \times H$ is the product of the orders of $K$ and $H$ . This means that if $|K| = m$ and $|H| = n$ , then $|K \times H| = mn$ .

Q: What is the relationship between the subgroups of a product group and the subgroups of its factors?

A: If $G = K \times H$ , then the subgroups of $G$ are of the form $K \times H'$ , where $H'$ is a subgroup of $H$ . Conversely, if $H'$ is a subgroup of $H$ , then $K \times H'$ is a subgroup of $G$ .

Q: Can a product group be written as a direct product of two smaller groups?

A: Yes, a product group $G = K \times H$ can be written as a direct product of two smaller groups if and only if $K$ and $H$ are both direct products of smaller groups.

Q: What is an example of a product group?

A: An example of a product group is the group of integers modulo $n$ , which can be written as a product group of the group of integers modulo $m$ and the group of integers modulo $k$ , where $n = mk$ .

Q: Can a product group be abelian?

A: Yes, a product group $G = K \times H$ can be abelian if and only if both $K$ and $H$ are abelian.

Q: Can a product group be simple?

A: No, a product group $G = K \times H$ cannot be simple, since it has non-trivial subgroups of the form $K \times \{e\}$ and $\{e\} \times H$ .

Q: Can a product group be finite?

A: Yes, a product group $G = K \times H$ can be finite if and only if both $K$ and $H$ are finite.

Q: Can a product group be infinite?

A: Yes, a product group $G = K \times H$ can be infinite if and only if at least one of $K$ or $H$ is infinite.

Q: Can a product group be cyclic?

A: No, a product group $G = K \times H$ cannot be cyclic, since it has non-trivial subgroups of the form $K \times \{e\}$ and $\{e\} \times H$ .

Q: Can a product group be solvable?

A: Yes, a product group $G = K \times H$ can be solvable if and only if both $K$ and $H$ are solvable.

Q: Can a product group be nilpotent?

A: Yes, a product group $G = K \times H$ can be nilpotent if and only if both $K$ and $H$ are nilpotent.