Let S be a subset of a group and let H be the intersection...

Let $S$ be a subset of a group and let $H$ be the intersection of all subgroups of $G$ that contain $S$. a. Prove that $\langle S\rangle=H$. b. If $S$ is nonempty, prove that $\langle S\rangle=\left\{s_{1}^{n_{1}} s_{2}^{H_{2}} \cdots s_{m}^{n_{m}} \mid m \geq 1, s_{i} \in S,\right.$, $\left.n_{i} \in Z\right\} .$ (The $s_{i}$ terms need not be distinct.)

Let $S$ be a subset of a group and let $H$ be the intersection of all subgroups of $G$ that contain $S$.
a. Prove that $\langle S\rangle=H$.
b. If $S$ is nonempty, prove that $\langle S\rangle=\left\{s_{1}^{n_{1}} s_{2}^{H_{2}} \cdots s_{m}^{n_{m}} \mid m \geq 1, s_{i} \in S,\right.$,
$\left.n_{i} \in Z\right\} .$ (The $s_{i}$ terms need not be distinct.)

Key Concepts

Subgroup Generated by a Subset

This concept involves constructing the smallest subgroup that contains a given subset of a group. It is defined as the intersection of all subgroups that contain the subset, ensuring that any subgroup containing the subset must also contain the generated subgroup, thereby capturing the minimal algebraic structure needed to include the subset.

Intersection of Subgroups

The intersection of subgroups is itself a subgroup. This key concept is used to show that the intersection of all subgroups containing a given subset is itself a subgroup, making it a natural candidate for the subgroup generated by the subset, because it includes all elements required by the subgroup properties and is minimal with that property.

Closure Properties in Groups

Understanding the subgroup generated by a subset requires knowledge of the closure properties of groups, which include closure under the group operation and the existence of inverses. These properties guarantee that any element in the generated subgroup can be written as a finite product of elements of the subset and their inverses, ensuring that the structure adheres to the group axioms.

Representation of Elements as Products of Powers

A central part of the discussion is that every element in the subgroup generated by a nonempty subset can be expressed as a finite product of elements from the subset raised to integer powers. This representation encapsulates both the inclusion of each element and its inverse, solidifying the concept that the subgroup generated by the subset is composed of all such possible combinations.

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