I'm trying to use Magma to do a double coset calculation on the group M10, but the answer does not make sense to me. Your help and comments are most appreciated. First, here's the calculation:


(1) M10 has one conjugacy class of order 72 subgroups; pick one and call it T. There is also one class of order 2 subgroups; pick one and call it S.


(2) Denote by $\tau\_i$ the representatives of the double coset decomposition
 $M10 = (T \tau\_1 S) \coprod (T \tau\_2 S) \coprod$ ...


(3) For each i: Compute the index $[ S^\tau\_i: T \cap S^\tau\_i ]$ where $S^t$ denotes the conjugate of S by t.


The (unordered) set of these indices is known to be independent of the choice of the tau's. However, when I tried to do this on Magma, I get different answers if I run the same code multiple times. I did run my code using smaller groups and I got the correct answer. I am new to Magma so perhaps I made syntax errors, but I'm totally confused. I've sent along my short code; your help and comments are most appreciative. **THANKS!**


=== magma code ===



```
G := SmallGroup(720,765);
S72  := Subgroups(G: OrderEqual:=72);
printf "There are %o class of index 10 subgroups\n",  #S72;
T   := S72[1]`subgroup;

S2  := Subgroups(G: OrderEqual:=2);
printf "There are %o class of order 2 subgroups\n",  #S2;
S  := S2[1]`subgroup;
printf "Here is the actual order 2 subgroup: %o\n", S;

printf ("Double coset representatives of the order 72 by order 2:  \n");
D2, D2size := DoubleCosetRepresentatives(G, T, S);
D2;

for i := 1 to #D2 do
  dtau := S^D2[i];
  printf "i=%o: %o\n", i, #dtau / #(T meet dtau);
end for;

```


Double coset representatives and Magma
The problem with inconsistent results in the final part of your calculation is not just due to different choices of random elements, as I said in my comment, but probably due to a confusion about the meaning of $S^t$, which is defined to be $t^{-1}St$ in Magma.


Think of the group $S$ as acting by right multiplication on the cosets $Tt$ of $Tt$ of $T$. Then the stabilizer of $Tt$ is $S \cap t^{-1}Tt$, and hence the length of the orbit of $S$, which is what you are trying to calculate, is $|S:S \cap t^{-1}Tt|$, which you could calculate as $|S|/|S \cap T^t|$ in Magma or, if you prefer, $|S^{t^{-1}}|/|S^{t^{-1}} \cap T|$.


So you could correct your code by replacing



```
dtau := S^D2[i]; 

```

by



```
dtau := S^(D2[i]^-1);

```

I tried that, and I got consistent results $1,1,2,2,2,2$ (not necessarily in that order) for the indices 


