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A group action of a group $G$ on a set $A$ is a map from $G \times A$ to $A$, written as $g \cdot a$, that satisfies two properties: compatibility with the group operation ($g_1 \cdot (g_2 \cdot a) = (g_1 g_2) \cdot a$) and the identity property ($1 \cdot a = a$).

Conditions: $G$ is a group.; $A$ is a set.; The action is a map from $G \times A$ to $A$.; Property (a) holds for all $g_1, g_2 \in G$ and $a \in A$.; Property (b) holds for all $a \in A$.

A group action of a group $G$ on a set $A$ is a map from $G \times A$ to $A$, written as $g \cdot a$, that satisfies two properties: compatibility with the group operation ($g_1 \cdot (g_2 \cdot a) = (g_1 g_2) \cdot a$) and the identity property ($1 \cdot a = a$).

Conditions: $G$ is a group.; $A$ is a set.; The action is a map from $G \times A$ to $A$.; Property (a) holds for all $g_1, g_2 \in G$ and $a \in A$.; Property (b) holds for all $a \in A$.

To verify if a map is a group action, first check that its domain is $G \times A$ and its codomain is $A$. Then, verify the two properties: $g_1 \cdot (g_2 \cdot a) = (g_1 g_2) \cdot a$ for all $g_1, g_2 \in G$ and $a \in A$, and $1 \cdot a = a$ for all $a \in A$.

Conditions: The map is defined from $G \times A$ to $A$.; $G$ is a group and $A$ is a set.

To verify if a map is a group action, first check that its domain is $G \times A$ and its codomain is $A$. Then, verify the two properties: $g_1 \cdot (g_2 \cdot a) = (g_1 g_2) \cdot a$ for all $g_1, g_2 \in G$ and $a \in A$, and $1 \cdot a = a$ for all $a \in A$.

Conditions: The map is defined from $G \times A$ to $A$.; $G$ is a group and $A$ is a set.

No, the set $A$ can be any arbitrary set and does not need to have any structural relationship with the group $G$.

Conditions: $G$ is a group and $A$ is a set.

No, the set $A$ can be any arbitrary set and does not need to have any structural relationship with the group $G$.

Conditions: $G$ is a group and $A$ is a set.