Question:Left right confusion

Q: '''Some places use a left action convention and some places use a right action convention. Are proofs given using these conventions equivalent? How can proofs be translated between the conventions?'''

A: The key fact is that every group is isomorphic to its opposite group via the inverse map, i.e., the inverse map establishes a bijection between a group and the group that would have been obtained by flipping the order of multiplication. This guarantees that interchanging left and right uniformly everywhere does not have an effect on the truth of statements about groups, where all variables are quantified existentially or universally.

An illustration is the fact that left and right coset spaces are naturally isomorphic. This natural isomorphism proceeds via the inverse map.

However, this is not to say that once we fix a particular convention, we can change it willy-nilly. In particular, it is not true that there is always an automorphism sending any element to its inverse. See inverse map is automorphism iff abelian, and also note that although there are many groups in which it is true that every element is automorphic to its inverse, there are also many groups where this is not true. The smallest counterexample group is the semidirect product of Z7 and Z3, where, for any element of order three, there is no automorphism sending it to its inverse. (Link to proof needed).