HallsofIvy said:For "[itex]H_1[/itex]", yes, any subgroup must contain the identity so if H contains only one member, it must be just the identity permutation, (1).
There are, in fact, 3 different subgroups of order 2: {e, (12)}, {e, (13)}, and {e, (23)}. There is the single subgroup of order 3: {e, (123), (132)}. Of course, the subgroup of order 6 is the entire group.
S3 is a mathematical notation for the symmetric group with three elements. It consists of all possible permutations of three distinct objects.
Subgroups are groups that are contained within a larger group. They have the same operations and properties as the larger group, but with a smaller set of elements.
Lagrange's Theorem states that the order of a subgroup must divide the order of the larger group. In other words, the number of elements in a subgroup must be a factor of the number of elements in the larger group.
S3 has 4 subgroups: the identity subgroup with 1 element, two subgroups with 2 elements each, and the full subgroup with 3 elements.
Since S3 has 6 elements, its subgroups must have orders that are factors of 6. This means that the subgroups can have 1, 2, 3, or 6 elements. Additionally, the subgroups must contain the identity element and be closed under the group operation.