A normal subgroup is a subgroup of a group that is invariant under conjugation by elements of the larger group. In other words, if a normal subgroup is conjugated by any element of the larger group, it remains within the subgroup.
The order of S3xS3, also known as the direct product of two copies of the symmetric group S3, is 9! = 362,880 elements. This is because each copy of S3 has 3! = 6 elements, and the direct product multiplies the number of elements in each group.
To find the normal subgroups of S3xS3, we can use the fact that the normal subgroups of a direct product are the direct products of normal subgroups of each individual group. In this case, the normal subgroups of S3 are the trivial subgroup {e} and the alternating group A3, and the normal subgroups of S3xS3 are the direct products of these two subgroups.
Normal subgroups are important in group theory because they allow us to define factor groups, which are groups formed by "factoring out" the normal subgroup from the larger group. These factor groups have many useful properties and can help us understand the structure of the larger group.
No, a subgroup cannot be both normal and non-normal at the same time. A subgroup is either invariant under conjugation by all elements of the larger group (normal) or it is not (non-normal). It cannot have both properties simultaneously.