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If $S_n$ denotes the group of all permutations over the set $\{1,2,...,n\}$ with respect to the composition of mappings and $H$ be a subgroup of $S_n$ containing the transposition $(1,2)$ and the $n$-cycle $(1,2,...,n)$ then show that $H = S_n$.

How can I solve it?Please give me a hint.

Arnab Chattopadhyay.
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1 Answers1

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Note that we have that:

$$(1,2,3,...,n)^r(1,2)(1,2,3,...,n)^{n-r} = (r+1,r+2)$$

where we have $n+1=1, n+2=2$. Now we have that:

$$(i,j) = (i,i+1)(i+1,i+2)\cdots(j-2,j-1)(j-1,j)(j-2,j-1)\cdots(i+1,i+2)(i,i+1)$$

Therefore every transposition is an element of $H$. Now prove that every element of $S_n$ is a product of transpositions to finish the proof.

Stefan4024
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  • Ok.got it.But it is tough to think.Isn't it? – Arnab Chattopadhyay. Oct 03 '16 at 13:59
  • @ArnabChatterjee. To think what? I don't get your point. – Stefan4024 Oct 03 '16 at 14:00
  • The representation of $(i.j)$. – Arnab Chattopadhyay. Oct 03 '16 at 14:01
  • @ArnabChatterjee It's a classic excerice in symmetric groups, so I guess one should be familiar with the proof. In the second part it's easy to see why and how it works. I mean every transposition just "pushes" one of the $i,j$ forward till we reach the other one. – Stefan4024 Oct 03 '16 at 14:05
  • please suggest a good book of symmetric group.Does Gallian work well? – Arnab Chattopadhyay. Oct 03 '16 at 14:07
  • @ArnabChatterjee. I haven't read the Galian's book, but for introductory level I like Herstein, Gilbert, Rotman, Fraleigh and Hall. These are some of the books I've read on this topic and as far as I remember they had something on Symmetric group, but not much advanced, nor too much on it. Anyway my recommendation would be to focus on group theory too and read some introductions in that field. – Stefan4024 Oct 03 '16 at 14:11