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$\langle x \rangle$ is the subgroup generated by $x$, which is the smallest subgroup of $G$ containing $x$. It is built by including $x$, its inverse $x^{-1}$, the identity $1$, and all integral powers of $x$ and $x^{-1}$ required for closure.

Conditions: $G$ is a group with operation $\times$.; $x \in G$.

$\langle x \rangle$ is the subgroup generated by $x$, which is the smallest subgroup of $G$ containing $x$. It is built by including $x$, its inverse $x^{-1}$, the identity $1$, and all integral powers of $x$ and $x^{-1}$ required for closure.

Conditions: $G$ is a group with operation $\times$.; $x \in G$.

A group is cyclic if it can be generated by a single element. In the video, this is written as $G = \langle x \rangle$, meaning every element of the group belongs to the subgroup generated by $x$.

Conditions: $G$ is a group.; There exists an element $x \in G$ such that $G = \langle x \rangle$.

A group is cyclic if it can be generated by a single element. In the video, this is written as $G = \langle x \rangle$, meaning every element of the group belongs to the subgroup generated by $x$.

Conditions: $G$ is a group.; There exists an element $x \in G$ such that $G = \langle x \rangle$.