It is well known that any finite group can be embedded in Symmetric group $S_n$, $GL(n,q)$ ($q=p^m$) for some $m,n,q\in \mathbb{N}$. Can we embed any finite group in $A_n$, or $SL(n,q)$ for some $n,q\in \mathbb{N}$?
Embedding of finite groups
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0you can also embed $G$ in $S_n$ then embed $S_n$ in $SL$ as $\{0,1\}$-permutation matrices. – 2011-03-15
3 Answers
Yes.
The symmetric group Sym(n) is generated by { (1,2), (2,3), …, (n−1,n) }. You can embed Sym(n) into Alt(n+2) as the group generated by { (1,2)(n+1,n+2), (2,3)(n+1,n+2), …, (n−1,n)(n+1,n+2) }. This embedding takes a permution π in Sym(n) and sends it to π⋅(n+1,n+2)sgn(π), where sgn(π)∈{0,1} is the parity of the permutation.
In other words, G ≤ Sym(n) ≤ Alt(n+2) embeds any group into a (slightly larger) alternating group.
The general linear group GL(n,q) embeds in the special linear group SL(n+1,q) using a determinant trick. We just add a new coordinate to cancel out the determinant of the matrix from GL(n,q) so the result lands in SL(n+1,q).
$\operatorname{GL}(n,q) \cong \left\{ \begin{bmatrix} A & 0 \\ 0 & 1/\det(A) \end{bmatrix} : A \in \operatorname{GL}(n,q) \right\} ≤ \operatorname{SL}(n+1,q)$
In other words, G ≤ GL(n,q) ≤ SL(n+1,q) embeds any group into a (slightly larger) special linear group.
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0Yay, I get it, thank you vvvveeeerrryyyy much for your kind support, and explanation. :* – 2013-12-12
Yes we can.
For $A_n$, we can embed the given group in some $S_{n-2}$ and then for the additional two elements choose the identity or the transposition according as the element of $S_{n-2}$ is even or odd.
For $SL(n,q)$, we can embed the given group in some $GL(n-1,q)$ and then choose the diagonal element in the additional row and column as the reciprocal of the determinant of the element of $GL(n-1,q)$.
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0Ah, much clearer now. – 2011-03-15
To your first question (embeddable within $A_n$) I think the answer is yes for obvious reasons: one can embed $S_n$ within $A_{n+2}$.
(Consider the subset of $A_{n+2}$ that stabilizes the first $n$ elements (as a set), it's obvious that this set will consist of all permutations of these elements, where it may or may not interchange the last two points, depending on the sign. For instance for $n=3$ we retrieve $S_3$ as $(1\ 2\ 3)$, $(1\ 2)(4\ 5)$, etc...)