判断对称群S₃的子集{(1),(12),(23),(13)}是否为子群

Hey there! Since you're just diving into group theory, let's walk through this question clearly—great way to build intuition for subgroup rules.

First, let's fix a key misconception you had: you don’t only need to check if subset elements multiply to something in S₃ (that’s always true because S₃ is a group, so its operation is closed). For a subset to qualify as a subgroup, three core conditions need to hold (since associativity is inherited from the parent group, we don’t have to recheck that):

• The subset must contain the identity element of the parent group (which your subset does: (1) is there)
• The subset must be closed under the group operation: multiplying any two elements from the subset must give another element inside the subset
• Every element in the subset must have its inverse element also present in the subset (your subset does satisfy this: each transposition is its own inverse, and the identity’s inverse is itself)

But here’s the catch: your subset fails the closure condition—let’s prove that with an example. Let’s compute the product of (12) and (13) (remember, permutation composition is usually done right-to-left: apply (13) first, then (12)):

• (13) sends 1→3, 3→1, 2→2
• Then (12) sends 3→3, 1→2, 2→1
• Combining these, the permutation (12)(13) maps 1→3, 3→2, 2→1—this is the 3-cycle (1 3 2), which is not in your subset {(1),(12),(23),(13)}.

Even quicker: we can use Lagrange’s Theorem, which says the order (size) of a subgroup must divide the order of the parent group. S₃ has 6 elements, your subset has 4 elements. Since 6 doesn’t divide 4, this subset can’t possibly be a subgroup—no need to even check closure!

So to wrap up: this subset isn’t a subgroup, because it fails closure (and Lagrange’s Theorem gives an immediate red flag too).

内容的提问来源于stack exchange，提问作者mcaiojethewo