Linked Questions

0
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0answers
48 views

How to find cyclic subgroups of $S_4$ with certain numbers of elements [duplicate]

In the group $S_n$ consisting of the set of all bijections from $\{1,2,...,n\}$ to $\{1,2,...,n\}$ with a binary operation $◦$ denoting composition of functions. I am asked to find cyclic subgroups of ...
0
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0answers
46 views

List the cyclic subgroups of S4? [duplicate]

I am trying to list all the cyclic subgroups of $S_4$, and also list two examples of proper non-cyclic subgroups of $S_4$. Here's the general idea of what I have for the list: $S_4$, $A_4$, $\langle ...
39
votes
2answers
8k views

Enumerating all subgroups of the symmetric group

Is there an efficient way to enumerate the unique subgroups of the symmetric group? Naïvely, for the symmetric group $S_n$ of order $\left | S_n \right | = n!$, there are $2^{n!}$ subsets of the group ...
16
votes
2answers
33k views

How many distinct subgroups does the symmetric group $S_3$ have?

How many distinct subgroups does the symmetric group $S_3$ have? So when I do this question, is there a general technique to use? I know that the order of a subgroup divides that of the group. But ...
3
votes
3answers
555 views

How to prove that $\mathrm{SL}_{2} (\mathbb F_{3})$ is not isomorphic to $S_{4}$?

How to prove that $\mathrm{SL}_{2} (\mathbb F_{3})$ is not isomorphic to $S_{4}$? They are both group of order $24$ and both groups have elements of order $2, 3$ and $4$. I don't know how to work from ...
0
votes
2answers
72 views

Find sub group of $S_{10}$ that is isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}_2$

I have the following question : Find sub group of $S_{10}$ that is isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ What I did: I took a sub group that contains only $4$ elements and it has to be a ...
3
votes
1answer
309 views

How to prove that if n divides 4! there is a subgroup of order n in S4.

Question as in title. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. My approach is rather bullish - I'm simply trying to find subgroups of order n for each factor. Subgroup of order 1: $<()>$ (...
0
votes
1answer
167 views

Cyclic subgroups in $S_4$

Question: Show that there are cyclic subgroups of order $1,2,3 \ \text{and} \ 4$ in $S_4$ but $S_4$ does not contain any cyclic subgroup of order $ \geq 5$. (Note: I suppose $S_4$ are all permutation ...
1
vote
2answers
159 views

Subgroups of $S_4$ of order 6

I have to find the subgroups of $S_4$ of order 6: <(12),(123)>={1,(12),(123),(132),(23),(13)} but how much are? maybe 4 : <(12),(124)>={1,(12),(124),(142),(24),(14)} ...
0
votes
1answer
143 views

Find All Subgroups in $S_4$ with Order $8$

See page 4 for the proof on the existence of a Sylow $p$-subgroup I followed the algorithm presented in the proof to build a subgroup with $8$ elements: $$(1), \, (1 2), \, (3 4), \, (1 2)(3 4), \, (...
0
votes
3answers
118 views

Let $A = \{1,2,3,4\}$. Number of functions $f :A \to A$, such that $f(f(x))=x~\forall x \in A$ , is.

Let $A = \{1,2,3,4\}$. Number of functions $f :A \to A$, such that $f(f(x))=x, \forall x \in A$ , is. For this , what I thought was 2 cases are possible. Either an element maps to itself or 2 ...
0
votes
1answer
98 views

Subgroups of $S_n $

I have to solve these exercises but I don't know how to do: Describe (if they exist) all subgroups of order 6 in $S_4$; Describe (if they exist) all subgroups of order 8, 10, 12, 15, 20, 24 in $S_5$. ...
1
vote
2answers
90 views

conceptual question with example: proving groups are isomorphic

The question is to prove $D_8$ and the subgroup of $S_4$ generated by $(1 2)$ and $(1 3)(2 4)$ are isomorphic. I was able to show that the relations for $D_{8}$ follow when we set $b = (1 2)(1 3)(2 4)...
1
vote
1answer
51 views

SubGroups of $S_4$ of order $8$ and $12$. Describe by hand

What's a "good" way of describing $S_4$ subgroups by hand? The following post gives a pretty detailed stategy but i can't figure out some tools that are used: How to enumerate subgroups of ...
0
votes
0answers
79 views

Find the number of conjugacy classes in $S_4$

So I know all the elements of $S_4=\{e,(12),(13),(14),(23),(24),(34),(12)(34),(13)(24),(12)(23),(123),(132),(124),(142),(134),(143),(234),(243),(1234),(1342),(1423),(1243),(1432),(1324)\}$ I also ...

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