Equivalence of definitions of group action
This article gives a proof/explanation of the equivalence of multiple definitions for the term group action
View a complete list of pages giving proofs of equivalence of definitions
The definitions we have to prove as equivalent
Definition as a map from a product
Here denotes the identity element.
Further, the group action is termed faithful or effective if for every non-identity element , there exists such that .
Definition as a homomorphism to a symmetric group
Further, the action is termed faithful or effective if is injective.
Proof of equivalence
The first definition implies the second
Suppose we are given a map satisfying the conditions:
We now construct the map . For every , is defined as the map:
Clearly, is a function from to , and is the identity map (by the second assumption on . The fact that is in for every holds because:
Hence is a left inverse for , and a similar argument shows that it is a right inverse. Thus, is an invertible map from to , hence an element of .
Finally the fact that is a homomorphism follows from the first condition:
Further, suppose is faithful. Consider any two distinct elements of . Consider . Since are distinct, is a non-identity element, so by faithfulness, there exists such that . Using the definition of group action, we obtain that while . In particular, and are permutations that differ on , hence are not equal. Thus, distinct elements of map to distinct permutations, and is thus injective.
The second definition implies the first
Given a homomorphism , the map is given by:
Let us check that satisfies both the specified conditions:
The latter is because a homomorphism of groups takes the identity element to the identity element.
Further, if is injective, then for any non-identity element , , hence is not the identity permutation. Thus, there exists such that does not fix . Thus yields , as desired.