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I have the following question on a homomorphism between symmetric groups and it has been really a pain. I know I am supposed to use induction but, I seem to miss something essential so if anybody could give me some help, or even the solution I would very much appreciate it! Here we are:

Let $f: S_k \to S_n$ be an injective homomorphism. Assume that $f$ takes the transposition $(1,2)$ to some transposition in $S_n$. Prove that there are $1 \leq a_1,a_2,\cdots,a_k\leq n$ such that $$f(i,i+1) = (a_i, a_{i+1})$$ for all $i =1,2,3,...,k-1$ and that furthermore, $|\{a_1,...,a_k\}| = k$, where $||$ denotes the cardinality.

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up vote 2 down vote accepted

Hint: Conjugation preserves the cycle type.

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