Let 𝐑^* denote the group of all nonzero real numbers under...

Let $\mathbf{R}^{*}$ denote the group of all nonzero real numbers under multiplication. Let $\mathbf{R}^{+}$ denote the group of positive real numbers under multiplication. Prove that $\mathbf{R}^{*}$ is the internal direct product of $\mathbf{R}^{+}$ and the subgroup $\{1,-1\}$.

Key Concepts

Subgroup of Order Two

The subgroup comprising the elements {1, -1} is a simple yet crucial subgroup of the nonzero real numbers, showing up as the sign component in decompositions. This subgroup is isomorphic to the cyclic group of order two and its trivial intersection with the subgroup of positive real numbers allows every nonzero real number to be uniquely expressed as the product of a positive number and a sign, fulfilling one of the key conditions for an internal direct product.

Internal Direct Product

In group theory, an internal direct product is a way to express a group as the product of two (or more) subgroups that intersect trivially, meaning their only common element is the identity element, and such that every element of the group can be uniquely written as a product of elements from each subgroup. This concept is useful for decomposing groups into simpler, well-understood parts and understanding the structure of the group overall.

Nonzero Real Numbers as a Group

The set of nonzero real numbers forms a group under multiplication. This group, often denoted as R* or ?×, has the property that every nonzero real number has a multiplicative inverse and this structure underlies various constructions and decompositions in algebra, providing a rich example of a continuous group.

Subgroup of Positive Real Numbers

The positive real numbers form a subgroup of the nonzero real numbers under multiplication. This subgroup is of particular interest because it is connected, excludes the non-positive elements, and is instrumental in many group decompositions, including the representation of elements as magnitudes (from the positive reals) and signs (from a subgroup of order two).

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