Stabiliser of a partition

Question

I'm trying to find the character table of $H$, which is defined as follows:

$$H:= \{\sigma \in S_6 \mid A_i\sigma = A_j \quad \forall i,j\}.$$

Here, $A_i :=\{2i-1,2i\}$ for $i=1,2,3$. In the question, $H$ is also referred as the `stabiliser of the partition $\{1,2\mid\mid 3,4,\mid\mid 5,6\}$'.

I'm not sure if the definition of $H$ makes sense (how can, for example, $A_1\sigma$ be equal to $A_1$, $A_2$ and $A_3$?).

I'm guessing that either:

• $\sigma$ maps $A_i$ to $A_i$ for each $i=1,2,3$, or
• $\sigma$ maps $A_i$ to some $A_j$, but not necessarily with $j=i$.

Are either of these a standard definition of a `stabiliser of a partition'?

Answer 1

You're right that that definition makes no sense. The intended definition (and standard meaning of "stabilizer of a partition") is your second guess: for each $i$, $\sigma$ maps $A_i$ to $A_j$ for some $j$. That is, the set $\{\sigma(A_1),\sigma(A_2),\sigma(A_3)\}$ is equal to the set $\{A_1,A_2,A_3\}$.