Find all numbers $n$ such that $S_7$ contains an element of order $n.$

Question

Find all numbers $n$ such that $S_7$ contains an element of order $n.$

Identity is the only element of order $1.$So $n=1$ is possible.

Case 1: Elements that can be written as a unique cycle of length$≥2$:Cycles of length 2 to 7 can exists whence the corresponding values of n are 2 to 7.

Case 2: Elements that can be written as a product of two disjoint cycles of length$≥2$:Here the possible elements are $σ_1 σ_2$ where $σ_1$ and $σ_2$ are disjoint cycles of length$≥2$,such that $4≤|σ_1 |+|σ_2 |≤7$. So in this case the possible values of n are {lcm(|σ_1 |,|σ_2 | ): σ_1 & σ_2 are disjoint cycles of length≥2 with 4 ≤|σ_1 |+|σ_2 |≤7}={lcm(n_1,n_2 ): n_1,n_2∈Z^+-{1} with 4≤n_1+n_2≤7} (Since there always exist disjoint cycles of length n_1 and n_2 such that n_1,n_2∈Z^+-{1} with 4≤n_1+n_2≤7)={2,6,4,10,3,12}.

Case 3: Elements that can be written as a product of three disjoint cycles of length≥2:Here the possible elements are σ_1 σ_2 σ_3 where σ_1 and σ_2 are disjoint cycles of length≥2,such that 6≤|σ_1 |+|σ_2 |+|σ_3 |≤7. So in this case the possible values of n are {lcm(|σ_1 |,|σ_2 |,|σ_3 | ): σ_1,σ_2,σ_3 are disjoint cycles of length≥2 with 6≤|σ_1 |+|σ_2 |+|σ_3 |≤7}={lcm(n_1,n_2,n_3 ): n_1,n_2,n_3∈Z^+-{1} with 4≤n_1+n_2+n_3≤7} (Since there always exist disjoint cycles of length n_1,n_2,n_3 with n_1,n_2,n_3∈Z^+-{1} with 4≤n_1+n_2+n_3≤7)={2,6}. We further note that there^' s no element in S_5 whose representation as a product of disjoint cycles contains 4 or more cycles of length≥2.Consequently the above three cases exhaust all the posibilites for the order of the elements of S_5.Thus n=1,2,3,4,5,6,7,10,12.

Am I correct?

Answer 1

I, by using a computational software in Group Theory called GAP, could checked that:

gap> e:=Elements(SymmetricGroup(7));;
 s:=DivisorsInt(Size(e));;
 Print("The Group does not contain any elements of order", ":", "  ");
 for n in s do if Size(Filtered(e,t->Order(t)=n))=0 then Print( "n=", 
     n,",  "); fi;
 od;

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The Group does not contain any elements of order:  n=8,  n=9,  n=14,  n=
15,  n=16,  n=18,  n=20,  n=21,  n=24,  n=28,  n=30,  n=35,  n=36,  n=40,  n=
42,  n=45,  n=48,  n=56,  n=60,  n=63,  n=70,  n=72,  n=80,  n=84,  n=90,  n=
105,  n=112,  n=120,  n=126,  n=140,  n=144,  n=168,  n=180,  n=210,  n=
240,  n=252,  n=280,  n=315,  n=336,  n=360,  n=420,  n=504,  n=560,  n=
630,  n=720,  n=840,  n=1008,  n=1260,  n=1680,  n=2520,  n=5040,