equivalent atoms Structural symmetry of a molecule can be defined mathematically as a permutation of the atoms that exchanges at least two atoms but does not change the molecule's structure. Two atoms then can be said to be structurally equivalent if there is a structural symmetry that takes one to the other. Thus, for example, all four hydrogen atoms of methane are structurally equivalent, because any permutation of them will preserve all the bonds of the molecule. Likewise, all six hydrogens of ethane () are structurally equivalent to each other, as are the two carbons; because any hydrogen can be switched with any other, either by a permutation that swaps just those two atoms, or by a permutation that swaps the two carbons and each hydrogen in one methyl group with a different hydrogen on the other methyl. Either operation preserves the structure of the molecule. That is the case also for the hydrogen atoms cyclopentane, allene, 2-butyne, hexamethylenetetramine, prismane, cubane, dodecahedrane, etc. On the other hand, the hydrogen atoms of propane are not all structurally equivalent. The six hydrogens attached to the first and third carbons are equivalent, as in ethane, and the two attached to the middle carbon are equivalent to each other; but there is no equivalence between these two equivalence classes. Symmetry and positional isomerism Structural equivalences between atoms of a parent molecule reduce the number of positional isomers that can be obtained by replacing those atoms for a different element or group. Thus, for example, the structural equivalence between the six hydrogens of ethane means that there is just one structural isomer of ethanol, not 6. The eight hydrogens of propane are partitioned into two structural equivalence classes (the six on the methyl groups, and the two on the central carbon); therefore there are only two positional isomers of propanol (1-propanol and 2-propanol). Likewise there are only two positional isomers of butanol, and three of pentanol or hexanol. Symmetry breaking by substitutions Once a substitution is made on a parent molecule, its structural symmetry is usually reduced, meaning that atoms that were formerly equivalent may no longer be so. Thus substitution of two or more equivalent atoms by the same element may generate more than one positional isomer. The classical example is the derivatives of benzene. Its six hydrogens are all structurally equivalent, and so are the six carbons; because the structure is not changed