# Questions tagged [symmetric-groups]

A symmetric group is a group consisting of all permutations of given finite set, with composition of permutations as the binary operation. Should be used with the (group-theory) tag.

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### If $m<n$, show that there is a $1$-$1$ mapping $F:S_m\rightarrow S_n$ such that $F(fg)=F(f)F(g)$ for all $f,g\in S_m$

question: If $m<n$, show that there is a $1$-$1$ mapping $F:S_m\rightarrow S_n$ such that $F(fg)=F(f)F(g)$ for all $f,g\in S_m$. Where $S_n$ stands for symmetric group of degree $n$ my approach: ...
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### Question about coloring a Cube [closed]

The vertices of a cube are numbered from $1$ to $8$. (a) What are all the elements of $S_8$ which correspond to symmetries of the cube? (b) How many ways the vertices of the cube can be coloured ...
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### Writing explicitly $(s^2-1)^2+(t^2-1)^2$ as a polynomial in $st$ and $s+t$?

Consider the symmetric polynomial $$P(s,t)=(s^2-1)^2+(t^2-1)^2.$$ How can we write $P$ as a polynomial in the variables $st,t+s$? The Fundamental theorem of symmetric polynomials implies this is ...
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### How do I show two pairs of elements of $S_n$ are conjugate by the same element?

Let $\alpha, \alpha’, \beta, \beta’$ be distinct non-identity elements of $S_n$. Suppose there exists $\tau \in S_n$ such that $\alpha’ = \tau \alpha \tau^{-1}$ and $\beta’ = \tau \beta \tau^{-1}$. ...
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### the intersection of two permutation groups

Let it be G= < a , b > a group formed by two permutations from S_10: a=(1 2 3 4)(5 6 7)(8 9 10) and b=(1 3)(2 4)(5 10)(6 8)(7 9) How can I know that the intersection of < a > and < b > is ...
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### Is there an isomorphism $\operatorname{Sym}(P) \cong Aut(P) \cong Aut (\mathbb{Z}_p)$, where $p$ is prime and $P$ is a group of order $p$? [closed]

And what about $Aut(P) \cong Aut (\mathbb{Z}_p)$
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### Find cosets of $H$ in $G$

Let $G$ = $\operatorname{Sym}(\{1,2,3,4\})$ and let $H = ⟨(1,2,3,4),(2,4)⟩$. Write out all the cosets of $H$ in $G$ So, I know that $G$ contains $4!= 24$ elements, because it's the permutation group....
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### Proving/Disproving (via formal proof) that the Alternating group $A_n$ is a subgroup of the Symmetric group $S_n$

intuitively this makes sense and i can conceptualize how this would work, but i struggle to formalize a proof to express what my ideas are... Henceforth, Let n be an element of $\mathbb Z^+$ and ...
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### Does $\{a,b,c,c^2\}$ generate the same group as $\{a,b,c\}$?

Is generated group by $\{a,b,c,c^2\}$ same as group generated by $\{a,b,c\}$? I think the answer is YES. But here is a paragraph of J. Wolf's Book: Let $\triangle_8$ denote the regular octahedron (...
### There exists a group element $\sigma\in S_7$ under some conditions?
It is given that $\sigma\in S_7$ where $S_7$ is a symmetric group. Do there exist an element $\sigma$ such that $\sigma^{20}=\sigma$ and $\sigma\ne e$ where $e$ is an identity element? My attempt: ...