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This question is derived (as want to derive the formula for below problem, and also general approach) from :

Compute the number of distinct actions of $C_m$ on set $X,$ s.t. $|X|= n= 2m+1.$

Let there be $i= n/m$ disjoint cycles, of length $m$, in a permutation of length $n$.

Then, the ways to obtain such permutations is given by:

$\frac{((nCm)(n-mCm)...(n-imCm))}{i}*2^i.$

Where the denominator is for ordering among the disjoint cycles.

While, the multiplier ($2^i$) is for two possible orderings in each.

If $n= 7,$ then have products of the form $(abc)(def),$ and the number of product of $3$-cycles given by:

$(((7C3)(4C3))/2) * 4 = $ $((7.5).2).4)= 35.8= 280.$

In case want to generalise for different values of $m,$ then need to find multiplier factor accordingly.

Say, if $m=4,$ have: $i=n/4$, multiplier is given by number of different ordering possible: $1234, 1243, 1324, 1342, 1423, 1432 = 4C2= 6.$

$((((7C4)/6).(3C3))/2) * 4 = $ $((7.2.5)= 70.$ Where divisor of $2$ is for orderings among the two cycles.

Similarly, want to find for $m=5,$ have: $i=n/5$, multiplier should be given by number of different ordering possible given by $5C2= 10$ if a general formula to find denominator were possible. But, instead, it is given by $18$ as given below:

$12345, 12354, 12435, 12453, 12534, 12543,$

$13245, 13254, 13452, 13425, 13524, 13542,$

$14235, 14253, 14325, 14352, 14523, 14532, $

Edit: precursor to this post.

jiten
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    You need to be clear about what is meant by "distinct actions". I think a lot of people would interpret "distinct actions" as meaning "inequivalent actions". For example, the two actions of $C_2 = \langle g \rangle$ on ${1,2,3}$ in which $g$ maps to $(1,2)$ and to $(2,3)$ are equivalent actions. – Derek Holt Jul 30 '22 at 08:06
  • @DerekHolt Kindly give some value for the question given, so that can get better idea of the approach used. If possible, some hints too. – jiten Jul 30 '22 at 08:09
  • @fitzcarraldo If could please give value for the given question. – jiten Jul 30 '22 at 08:28
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    There are $350$ elements of order $3$ in $S_7$, making $351$ possible actions. But the number of inequivalent actions is just $3$. – Derek Holt Jul 30 '22 at 11:37
  • @DerekHolt Request to list the smallish set of size $3$ of inequivalent actions. Also, $350= 280 (70\times 4)+70$, so need add extra $70$ elements? Seems $70$ actions aren't counted in the process, or ignored. Also, what is one action not listed? – jiten Jul 30 '22 at 13:17

1 Answers1

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Remember that an action is a homomorphism from $C_3$ into $S_7$, here. (A group action on a set $X$ always gives a homomorphism from $G$ into $\rm{Sym}(X)$.)

The $3$ inequivalent actions are $1\to e,\,1\to(123)$, and $1\to(123)(456)$, for instance.

There are $7\cdot 6\cdot 5/3=70$ three cycles.

And there are $70\cdot 4\cdot 3\cdot 2/(3\cdot 2)=280$ products of disjoint three cycles.

So $350$ elements of order three.

$1$ in $C_3$ can go to any of these, or to $e$. That's $351$ actions (total).

calc ll
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  • Thanks a lot, though request the first paragraph's elaboration too. – jiten Jul 30 '22 at 13:52
  • Why "$1$"? Homomorphisms send identity to idenity. –  Jul 30 '22 at 14:08
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    $1$ isn't the identity; it's a generator. @fitzcarraldo – calc ll Jul 30 '22 at 14:11
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    Ah ok, so you think additively, say $\Bbb Z_3$, while I thought multiplicatively, say ${e,a,a^2}$. –  Jul 30 '22 at 14:14
  • What will be the permutation corresponding to that. – jiten Jul 30 '22 at 14:15
  • Kindly find my doubts. By homomorphism: An action $\varphi:C_3\to S_X$ is fully determined by $\varphi(g)$ where $C_3=\langle g\rangle.$ Then if $g$ acts by $\sigma,$ then $g^2$ acts by $\sigma^2$ and left option is for $e$ to map to identity.

    Doubt#1: What is the permutation map of $e$ to identity, an actual value set?

    Doubt#2: You differ in mapping $e$ to any of $350$ choices too?

     – jiten Jul 30 '22 at 14:48
  • Have a similar question, but number of mappings are much smaller. Hope that can help learn better: Compute the number of distinct actions of $C_6$ on $X={1,2,3,4,5,6,7}$. $\varphi: C_6\rightarrow S_X$ is fully determined by $\varphi (g)$ where $C_6=\langle g\rangle$. $(7C 6)\times$ #of orderings ($=1$) of elements per selection.

    Doubt#1: Am unclear what adding $e\rightarrow e$ means to it. Or, how it gives a permutation to add to.

    Anyway, by following your answer it should be: $8$?

     – jiten Jul 30 '22 at 15:11
  • Kindly ignore my doubts for identity map, as they are now obvious by definition of action of a group on a set. – jiten Jul 31 '22 at 00:29
  • Kindly see related post: https://math.stackexchange.com/q/4503517/424260. – jiten Jul 31 '22 at 13:26
  • Kindly vet my answer (made at @StinkingBishop's invitation) at: https://math.stackexchange.com/a/4504206/424260 – jiten Aug 01 '22 at 22:29
  • Kindly help with post at: https://math.stackexchange.com/q/4504530/424260. – jiten Aug 02 '22 at 06:41