In abstract algebra an inner automorphism is a function in which an initial operation is applied, then another operation, and then the initial operation is reversed. With letters to indicate the operations and thing-being-transformed: f∘g∘f(X). Sometimes the initial action and its subsequent reversal change the overall result ("raise umbrella, walk through rain, lower umbrella" has a different result from just "walk through rain"), and sometimes they do not ("take off left glove, take off right glove, put on left glove" has the same effect as only "take off right glove").