Generators of the Multiplicative Group of Nonzero Real Numbers

The multiplicative group of nonzero real numbers, denoted as ( mathbb{R}^* ), encompasses all real numbers except zero under multiplication. This group is of significant interest in algebra and analysis, particularly in understanding the structure and properties of real numbers.

Understanding the Multiplicative Group

The multiplicative group of nonzero real numbers ( mathbb{R}^* ) is a group where multiplication is the binary operation. This group is uncountable and plays a crucial role in various mathematical contexts. One of the key questions in group theory is to identify a set of generators for the group, i.e., a subset of ( mathbb{R}^* ) from which every element in ( mathbb{R}^* ) can be generated.

Generators and Subsets of ( mathbb{R}^* )

To find a set of generators for ( mathbb{R}^* ), we can consider the following:

Positive Real Numbers

The positive real numbers, denoted ( mathbb{R}^ ), form a subgroup of ( mathbb{R}^* ). Importantly, the positive real numbers can be generated by the single element ( e ), the base of the natural logarithm, approximately 2.718. This is due to the fact that any positive real number ( x ) can be expressed as ( e^y ) for some ( y in mathbb{R} ).

Mathematically, every positive real number can be written in the form ( e^y ), where ( y ) is a real number.

Negative Real Numbers

The negative real numbers can also be generated using a single element: the number ( -1 ). Every negative real number can be expressed as the product of ( -1 ) and a positive real number. Thus, the negative real numbers can be represented as ( -1 cdot r ), where ( r ) is a positive real number.

Combining these observations, we can conclude that the multiplicative group of nonzero real numbers ( mathbb{R}^* ) can be generated by the set ( { -1, e } ).

Minimality and Uncountability

The minimality of a generating set means that no proper subset of the generating set can generate the entire group. For ( mathbb{R}^* ), the set ( { -1, e } ) is not minimal because the subgroup generated by either ( -1 ) or ( e ) alone can generate the whole group. Therefore, there is no minimal generating set for ( mathbb{R}^* ).

Uncountability and Generating Sets

Since the multiplicative group ( mathbb{R}^* ) is uncountable, it cannot be finitely generated nor even countably generated. Consequently, any generating set for ( mathbb{R}^* ) must be uncountable. There are uncountably many generating sets available for ( mathbb{R}^* ), which includes sets such as:

The real numbers except the primes ( {2, 3, 5, 7, ldots} ). The irrational numbers. The transcendental numbers. The numbers that are within one trillionth of ( -1 ) or ( -23 ). The positive numbers whose natural logarithms have decimal expansions consisting entirely of 0s and 1s.

These sets are all valid generators for the multiplicative group ( mathbb{R}^* ).

Dissociation of Minimality and Divisibility

In the context of group theory, a group is divisible if for every element in the group and every positive integer ( n ), there is an element in the group such that its ( n )-th multiple equals the original element. For instance, the additive group of real numbers ( mathbb{R} ) is divisible but does not have a minimal generating set. This is because the additive group of positive reals ( mathbb{R}_ ^* ) doesn't have a minimal generating set either; these groups are isomorphic under the exponential map.

Conversely, the multiplicative group of all nonzero reals ( mathbb{R}^* ) is not a divisible group. It is the direct product of a cyclic group of order 2 (which contains ( -1 )) and the group of positive reals ( mathbb{R}_ ^* ). If ( mathbb{R}^* ) had a minimal generating set ( S ), we could choose a finite subset ( A ) of ( S ) such that ( -1 ) is generated by ( A ). However, this leads to a contradiction because ( mathbb{R}^*/langle A rangle ) would be a divisible group with a minimal generating set, which is not possible for a direct product of a cyclic group and a divisible group.

Conclusion

In summary, the multiplicative group of nonzero real numbers ( mathbb{R}^* ) does not have a minimal generating set. This is due to its uncountability and the fact that it cannot be finitely or countably generated. Furthermore, the group does not possess a minimal generating set because of its structure and properties in relation to divisibility and direct products.

Understanding the generators and properties of groups like ( mathbb{R}^* ) is crucial for advanced mathematics, including algebra, analysis, and number theory.