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I'm being asked to find 2 subgroups of $S_3$, one of which is normal and one that isn't normal. I guess, to find the non normal subgroup is easier.

I would do this by trial and error, but since the group is $S_3$ and easily visualizable, I guess that there should be a geometrical property that makes it easy to find a normal subgroup. Any of you guys know one? Or should I try this by brute force?

user26857
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Poperton
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  • The/a normal one is the alternating group $A_3$, I guess. I think this is the only normal subgroup other than {$e$}. Now you can look at a small mount of possibilities. – Gary. Jul 14 '15 at 23:32
  • For relatively small symmetric groups it's convenient to find an explicit form of their subgroups. For example, $S_{3}$ has 4 subgroups up to an isomorphism: trivail group, $S_{2}$, $A_{3}$, $S_{3}$ by itself. Then, it's straightforward to check, whether the group is normal or not. – hyperkahler Jul 14 '15 at 23:35
  • $S_3$ has only four proper nonidentity subgroups. One is normal and three are not. Trial and error really isn't bad. – Matt Samuel Jul 14 '15 at 23:36

6 Answers6

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HINT: Keep in mind that $|S_3|=6$, so if you have any subgroup of order 3, it has index 2 and is therefore normal. Can you find a subgroup of order 3?

To find a nonnormal subgroup, you will need a different divisor of 6 that is not trivial, and the only choice there is $2$, so you are looking for a 2 element subgroup. Can you find that?

Alan
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$S_3 =\{1 , (12),(13),(23),(123),(132)\}$,then $A_3 =\{1 , (123),(132)\}$ is normal subgroup of $S_3$ and $\{1 , (12)\} ,\{1,(13)\} ,\{1,(23)\} $ are non normal subgroups of $S_3.$

Math
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$S_3$ is small enough that one can solve this by brute force, comparing left and right cosets of every subgroup. My suggestion: calculate all the left and right cosets of $H_1 = \{e, (1\ 2)\}$ and $H_2 = \{e, (1\ 2\ 3),(1\ 3\ 2)\}$, first.

David Wheeler
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$H_1=\{e,(1 2)\}$ is not a normal subgroup of $S_3$. However, $H_2=\{e,(1 2 3),(1 3 2)\}$ is a normal subgroup of $S_3$.

Parcly Taxel
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따라 오세요
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    I do not see what this adds to the other answers from 6 years ago. In particular, it is covered by the answer of Math. (You would be better focusing on newer questions.) – user1729 May 18 '21 at 16:16
  • ı studied this topic today,so I just want to add this examples. thanks for advice :) I'm new using this website.(my english is bad ,sorry) – 따라 오세요 May 18 '21 at 16:22
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    A good rule of thumb is: do not rewrite already existing answers. Similarly, when asking about homework, show your own work, instead of trying to make others do it for you... – Jonas Linssen May 18 '21 at 17:06
  • thank you for advice @PrudiiArca:) – 따라 오세요 May 18 '21 at 17:07
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Normal subgroup of $S_3$ are $\{e\}$, $\{e, (123), (132)\}$, and $S_3$ itself. All subgroups with two elements are not normal.

user26857
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jason
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You can do it using Sylow theory, notice that any non trivial proper subgroup of $S_3$ is of order 2 or 3, so they must be sylow groups of order 2 or 3, in this case $n_3=1$ and $n_2=3$, so we a have only one normal subgroup of order 3, and 3 subgroups of order 2 that are not normal.

Uriel Guerrero Valadez
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    Please don't answer old questions that already have accepted answers. In this case the Sylow theorems are substantially more advanced than what's needed, or helpful to a beginner. – Ethan Bolker Jan 01 '25 at 21:51
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    I understand what you say, and i know you can do it by directly calculation, however is interesting to see another way to solve something, and apart from that everybody is free to answer. – Uriel Guerrero Valadez Jan 01 '25 at 21:56