Help with the transposition proof?

Be careful

a transposition. The signum is defined as

where inv(τ) is the amount of the faults. Each pair of two numbers, which is “false” after the permutation, is a mistake. If, for example, the 5 is left of the 2, after the permutation, (2, 5) is a defect.

In a transposition, we have an obvious maladministration, namely (i, j). In the natural order i stood before j, but the transposition has changed i and j. If the transposition is for example (25), then 5 is the second element and 2 is the fifth. However, it is 2 < 5, i.e. 2 should be left of the 5 - this is the fault. Inv(τ) thus contains exactly one element, i.e.:

In principle, you have to go through all (ordered) number pairs during the manual calculation of the signum – all. In the symmetrical group of order 5, these would be:

Then check for each number pair whether it is still in this order after permutation or it has been mirrored. If it was mirrored, it’s a mistake. At the end, you count all the maladministrations together – there are just many, the Signum 1, are odd many, is the Signum -1.

An alternative is the product formula for the signum, this reduces the manual effort, but is usually associated with much more computational effort.