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Any subgroup containing $x$ must also contain its inverse $x^{-1}$ and the identity element $1$ because these are required by the definition of a subgroup. Additionally, closure under the group operation forces the inclusion of all integral powers of $x$ and $x^{-1}$.

Conditions: $G$ is a group.; $H \le G$ is a subgroup.; $x \in H$.

Any subgroup containing $x$ must also contain its inverse $x^{-1}$ and the identity element $1$ because these are required by the definition of a subgroup. Additionally, closure under the group operation forces the inclusion of all integral powers of $x$ and $x^{-1}$.

Conditions: $G$ is a group.; $H \le G$ is a subgroup.; $x \in H$.

No, writing $+$ or $\times$ does not mean the group elements are numbers. These are abstract notational conventions.

Conditions: Used for abstract groups.; The choice of notation changes the symbols for operation and identity but not the underlying group structure.

No, writing $+$ or $\times$ does not mean the group elements are numbers. These are abstract notational conventions.

Conditions: Used for abstract groups.; The choice of notation changes the symbols for operation and identity but not the underlying group structure.