All Subgroups of S3: Lagrange's Theorem Explained

[e179285]

The question wants all subgroups of S₃ . If H≤S ₃ , then ; IHI=1,2,3,6 by Lagrance's Theorem.

In other words, order of H can be 1,2,3 and 6.

What ı want to ask is how to write subgroup of S₃. For example,is H ₁ (1) ?

 

[HallsofIvy]

For "$H_1$", 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.

 

[e179285]

For "$H_1$", 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.

What can H₅ be?

 

[HallsofIvy]

? You just said that every subgroup of S3 (every subgroup of any group of order 6) must have order 1, 2, 3, or 6 (a divisor of 6). What do you mean by "H₅"?