置换性质：对换——$ au_{ij}= au_{ji}$的含义是什么？

Great question—this is one of those foundational permutation properties that clicks once you break down what the notation actually means. Let's walk through it step by step.

First, let's recap what a transposition $ au_{ij}$ is: it's a permutation of the set ${1, 2, ..., n}$ that only swaps the elements $i$ and $j$, leaving every other element exactly where it is. No other elements get moved—this is a simple two-element swap.

Now, the equality $ au_{ij} = au_{ji}$ is just stating that swapping element $i$ with element $j$ is identical to swapping element $j$ with element $i$. Let's use a concrete example to make this tangible:

Suppose we're working with permutations of 5 elements. $ au_{24}$ takes the sequence $[1, 2, 3, 4, 5]$ and transforms it into $[1, 4, 3, 2, 5]$. What does $ au_{42}$ do? It swaps 4 and 2—applying it to the same starting sequence also gives $[1, 4, 3, 2, 5]$. There's zero difference between the two operations.

To formalize this a bit (without overcomplicating):

• For any element $k$ that's not $i$ or $j$, both $ au_{ij}(k)$ and $ au_{ji}(k)$ return $k$—neither permutation touches these elements.
• For element $i$: $ au_{ij}(i) = j$, and $ au_{ji}(i)$ is also $j$ (since $ au_{ji}$ swaps $j$ and $i$, it maps $i$ directly to $j$).
• For element $j$: $ au_{ij}(j) = i$, and $ au_{ji}(j)$ is also $i$ (swapping $j$ and $i$ maps $j$ right back to $i$).

Since two permutations are equal if and only if they act exactly the same way on every element in the set, this means $ au_{ij}$ and $ au_{ji}$ are the exact same permutation.

The notation can trip you up at first—you might think the order of $i$ and $j$ matters—but it's just a labeling convention. Whether you write the first index as $i$ or $j$, you're referring to the same swap operation. It's like saying "swap the cat and dog" is the same as "swap the dog and cat"—the order you mention them doesn't change what you're doing.

内容的提问来源于stack exchange，提问作者Anonymus