Conjugacy class of transpositions is preserved by automorphisms
For the symmetric group on a finite set
Suppose is a finite set of size greater than and not equal to and is the symmetric group on . Then, the conjugacy class of transpositions in is preserved by automorphisms. In other words, any automorphism of sends transpositions to transpositions.
For the symmetric group on an infinite set
Suppose is an infinite set and is the symmetric group on . Then, the conjugacy class of transpositions in is preserved by automorphisms of . In other words, every automorphism of sends transpositions to transpositions.
- Conjugacy class of transpositions is the unique smallest conjugacy class of involutions: This statement holds when the size of the set is not equal to .
- Finitary alternating group is monolith in symmetric group
A proof that works only for the finite case
For the finite case, fact (1) gives the proof. This is because any automorphism preserves the order of elements, and preserves conjugacy, hence it permutes the conjugacy classes of involutions (i.e., of elements of order two). Since the conjugacy class of transpositions is the unique smallest conjugacy class of involutions, it cannot be mapped to another conjugacy class of involutions, hence it remains invariant under automorphisms.
A proof that works in general
Given: A set of size greater than and not equal to (possibly infinite). is the symmetric group on . A transposition of and an involution of that is not a transposition.
To prove: There is no automorphism of sending to .
Proof: If there were an automorphism of sending to it would send to , so the two groups would be isomorphic, and hence, the largest normal -subgroup of is isomorphic to the largest normal -subgroup of . (By a -subgroup, we mean a subgroup where all the elements have order a power of ).
Suppose and is the complement of . Then, .
We first consider the case that has at least seven elements, so that has at least five elements.
For finite fact (2) yields that if has five or more elements or is infinite, then the finitary alternating group on is contained in every nontrivial normal subgroup of . From this, we see that has no nontrivial normal -subgroup. Thus, the largest normal -subgroup of is .
On the other hand, is a product of (possibly infinitely many) disjoint transpositions. The subgroup generated by all these transpositions is a normal -subgroup in . Since is not a transposition, it involves at least two transpositions, so has a normal -subgroup of order at least four. Thus, the largest normal -subgroup of is not isomorphic to that of .
We now consider the case that has fewer than six elements. If has two elements, three elements, or five elements, the largest normal -subgroup of is still trivial, so the above applies.
When has four elements, has two elements, and in this case, is a Klein four-group. The only other possible conjugacy class of involutions is double transpositions, and for a double transposition , is isomorphic to the dihedral group of order eight.