# properties of alternating subgroup?

I was wondering, is it true that if $Alt_n$ is an alternating subgroup of $Sym_n$ for $n>3$, $Alt_{n-i}\leq Alt_n$ for all $i<n$?

• What do you mean by $\le$ here? Does that mean that $Alt_{n-i}$ is a subgroup of $Alt_n$?
– MJD
Mar 26 '15 at 16:20
• Yes! I do mean that. Mar 26 '15 at 16:21
• More interesting is, that $S_n$ cannot be embedded into $A_{n+1}$, see here. Mar 26 '15 at 16:24

The answer is no, in a strict sense. A permutation of $\{1,2,3\}$ is not a permutation of $\{1,2,3,4\}$. However, there is a natural bijection between the set of permutations of the former, and the set of permutations of the latter that fix $4$. Similarly, there is a natural bijection between $Alt_{n-i}$ and a subgroup of $Alt_n$, namely those permutations that fix $(n-i+1), (n-i+2),\ldots, n$.
• I guess that is what I was looking for then... I meant, is it true then that there exists a subgroup of $Alt_{n}$ that is isomorphic to $Alt_{n-i}$ for all $i<n$? Mar 26 '15 at 16:25
• It is clear, but it should still be stated, that every permutation in $Alt_n$ in the range of the bijection is an even permutation: i.e. it really is in $Alt_n$ and not just in $Sym_n$. Mar 26 '15 at 16:30
• In fact, there are several such isomorphisms besides the "obvious" one, as you can make $n$ distinct isomorphs of $S_{n-1}$ inside $S_n$, by choosing different elements to be "fixed". Mar 26 '15 at 19:20