# Thread: Symmetric Group Subgroups

1. ## Re: Symmetric Group Subgroups

and how did you arrive at that conclusion? it's not that i don't believe you, but rather, in mathematics, the pudding is in the proof.

2. ## Re: Symmetric Group Subgroups

I tested every element and saw gxg^-1 was always in <x>. Then I stopped because you said there was only 1! But I see that I could find counterexamples for the other subgroups to show they are not.

What I still don't really understand is how to find the subgroups in the first place.

3. ## Re: Symmetric Group Subgroups

this is where lagrange's theorem comes in. besides S3 and 1, you can only have subgroups of order 2 and 3.

every element of order 2 generates a subgroup of order 2. S3 has 3 elements of order 2, hence 3 subgroups of order 2. see?

every element of order 3 generates a subgroup of order 3. S3 has two elements of order 3, both of which generate the same subgroup.

now, 2 and 3 are prime, so subgroups of prime order can only be made from a generator. we've found all the generators, so we've found all the subgroups

(this is no longer true in larger groups, be thankful S3 is so nice).

4. ## Re: Symmetric Group Subgroups

Looks like I need to go study up on that theorem. Thanks a lot for all your help! You cleared up a lot for me.

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