# 3-dimensional array

I apologize if my question is ill posed as I am trying to grasp this material and poor choice of tagging such question. At the moment, I am taking an independent studies math class at my school. This is not a homework question, but to help further my understanding in this area. I've been looking around on the internet for some understanding on the topic of higher dimensional array. I'm trying to see what the analogue of transposing a matrix is in 3-dimensional array version.

To explicitly state my question, can you transpose a 3-dimensional array? What does it look like? I know for the 2d version, you just swap the indices. For example, given a matrix $A$, entry $a_{ij}$ is sent to $a_{ji}$, vice versa. I'm trying to understand this first to then answer a question on trying to find a basis for the space of $n \times n \times n$ symmetric array.

There is no single transformation corresponding to taking the transpose. The reason is that while there is only one non-identity permutation of a pair of indices, there are five non-identity permutations of three indices. There are two that leave none of the fixed: one takes $a_{ijk}$ to $a_{jki}$, the other to $a_{kij}$. Exactly what permutations will matter for your underlying question depends on how you’re defining symmetric for a three-dimensional array.