According to Wikipedia ("Symmetric group", 09-Mar-2019, https://en.wikipedia.org/wiki/symmetric_group) "In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions." and according to Encyclopedia of Mathematics ("Symmetric group", 09-Mar-2019, https://www.encyclopediaofmath.org/index.php/Symmetric_group) "The group of all permutations (self-bijections) of a set with the operation of composition (see Permutation group).". In [Rotman] p. 27 "If X is a nonempty set, a permutation of X is a function a : X -> X that is a one-to-one correspondence." and "If X is a nonempty set, the symmetric group on X, denoted S<sub>X</sub>, is the group whose elements are the permutations of X and whose binary operation is composition of functions.". Therefore, we define the <b>symmetric group</b> on a set as the set of one-to-one onto functions from to itself under function composition, see df-symg. However, the set is allowed to be empty, see symgbas0. <i>Hint: The symmetric groups should not be confused with "symmetry groups" which is a different topic in group theory.</i> <br><br> In this context, the one-to-one onto functions are called <b>permutations</b> for short. Since the base set of symmetric groups on a set is the set of all permutations of (see symgbas), we can formally say expressing " is a permutation of " if we are not interested in the group (or topology) structure. <br><br> In general, a <b>permutation group</b> "... is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself)." (see Wikipedia "Permutation group", 17-Mar-2019, https://en.wikipedia.org/wiki/Permutation_group). This means that a symmetric group is a permutation group, and each permutation group is a subgroup of a symmetric group (see pgrpsubgsymgbi and pgrpsubgsymg). For example, the structure with the singleton containing only the identity function restricted to a set as base set and the function composition as group operation is a permutation group (group consisting of permutations), see idrespermg, which is a (proper) subgroup of a symmetric group, see idressubgsymg. <br><br> As in [Rotman] p. 28 "Let and ; we say <b>fixes</b> if ; otherwise <b>moves</b> .". The theorems starting with symgfix2 are about fixed/moved elements.