Finding a normal and not normal subgroup of $S_3$

Question

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?

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

score 11 · Accepted Answer · answered Jul 14 '15 at 23:33

11

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?

answered Jul 14 '15 at 23:33

Alan

16,582

3

I was initially confused as to why index 2 implies it is normal. For anyone else, this question might help. – NNNComplex Aug 05 '21 at 14:51

score 7 · Answer 2 · answered Jul 15 '15 at 00:24

7

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

answered Jul 15 '15 at 00:24

Math

688

Hi why is {1,(12)},{1,(13)},{1,(23)} not Normal? I am new to Group theory so I am struggling a bit. – Dominic Joseph Feb 07 '22 at 09:29
@DominicJoseph they are not normal because they are not closed under conjugation. For example, $(123)(12)(123)^{-1}=(23)$ – Nothing special Feb 26 '24 at 18:50

score 3 · Answer 3 · answered Jul 15 '15 at 02:33

3

$S_3$ 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.

answered Jul 15 '15 at 02:33

David Wheeler

15,406

score 0 · Answer 4 · edited May 18 '21 at 14:15

0

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

edited May 18 '21 at 14:15

Parcly Taxel

103,344

answered May 18 '21 at 13:50

따라 오세요

19

1

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

score 0 · Answer 5 · edited Jan 25 '22 at 15:39

0

Normal subgroup of $S_3$ are $\{e\}$, $\{e, (123), (132)\}$, and $S_3$ itself. All subgroups with two elements are not normal.

edited Jan 25 '22 at 15:39

user26857

52,094

answered Jan 25 '22 at 14:23

jason

1

did you mean "subgroup with two elements"? – cineel Jan 25 '22 at 14:31
sorry for the mistake , you are correct. thank you for pointing – jason Jan 26 '22 at 14:23

Finding a normal and not normal subgroup of $S_3$

5 Answers5

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