Find the centralizer of $(123)$ in $S_6$.

Question

Find the centralizer of $(123)$ in $S_6$.

Is there any software to calculate $C_{(123)}$ where $C_{(123)}$ denotes centralizer of $(123)$ in $S_6$.

Can anyone say how to write the code to find centralizer of $(123)$ in $S_6$?

Can it be done using SageMath?

I checked that it can be done by SageMath but it only lists the generators of the centralizer of $(123)$ in $S_6$.

Is there any code to list all the elements of the centralizer of $(123)$ in $S_6$?

Answer 1

The centralizer of $a = (1,2,3)$ is rather simple to describe. First assemble the permutations of $\{4,5,6\}$. There are $6$ of them. Then precede each of these with the permutations $a^0,a,a^3$ which gives you the $18$ elements of the centralizer. Here is the code in GAP:

gap> c1 := SymmetricGroup([4..6]);
Sym( [ 4 .. 6 ] )
gap> c2 := Group((1,2,3));
Group([ (1,2,3) ])
gap> c := Group(Union(c1,c2));
Group([ (), (5,6), (4,5), (4,5,6), (4,6,5), (4,6), (1,2,3), (1,3,2) ])

But Gap has the direct command 'Centralizer':

gap> S6 := SymmetricGroup(6);
Sym( [ 1 .. 6 ] )
gap> Centralizer(S6,(1,2,3)) = c;
true

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