Good morning to you all! I have math question. Please help me out!
We take a subset $H$ from a symmetric group $S_n$ (it has n! elements, based on all permutations of a set of n distinct elements, such as the positive integers from 1 to n). Now consider $B(H)$ composed of all objects that can be written in the form $a * b$ where $*$ is the group operation of symmetric group and $a, b \in H$ ($a, b$ are elements of $H$). $$B(H)=\{g| g=a*b, \text{where} \quad a,b \in H\}$$
What sufficient conditions need to be imposed on set $H$ so $a * b \neq c*d$ for any $a,b,c,d \in H$ ?
PS:
$a=b$ is allowed so, $a*a \in B(H)$.
Note, we are composing $B(H)$ using exactly two elements from $H$ under symmetric group operation.
Of course, we are not considering trivial cases like when the set has only one element, e.g. $a=b=c=d$.
You are welcome to give examples and counter-examples.
If we adopt my suggestion of looking for sufficient conditions to be imposed on H in order to obtain the following ...
$$\forall a, b, c, d \in H [ [ a * b = c * d ] \Rightarrow [a = c \land b = d ] ]$$
... then it seems that the * operation is like formation of ordered pairs. That suggests that you might require that there be an isomorphism between the Cartesian product H x H and some subgroup of $S_n$. Does that idea have any value? The idea is that imposing that requirement could provide some information about what restrictions are imposed on H.
We can create an operation on the set H x H that makes (H x H, that operation) into a group by simply using the operation * on corresponding coordinate values: (r, s) * (t, u) = (r * t, s * u).
I'm just going to point out that in hoping to obtain $a * b \neq c*d$, you are using the operation * of the symmetric group $S_n$. That's why it is specifically a subgroup of $S_n$ that is to be isomorphic to H x H. Earlier, I edited to change what I had originally written to say that we want "an isomorphism between the Cartesian product H x H and some subgroup of $S_n$ x $S_n$." However, I changed it back, and thought it worthwhile to indicate why the original idea is a bit more subtle than it seems, and why a particular attempt to revise it would be a mistake.