Exponent of the alternating group.

Question

The exponent of a group $G$, denoted $\text{Exp}(G)$, is the smallest $n\in \mathbb{N}$ such that $x^n=1$ for every $x\in G$.

This page says that $$\text{Exp}(A_n)=\left\{\begin{array}{lcl}\text{lcm}\{1,\ldots,n-1\}&:&n \text{ is even}\\\text{lcm}\{1,\ldots,n\}&:&n \text{ is odd} \end{array}\right.$$

But I think this must be false. $A_3$ is just the cyclic group $C_3$ so its exponent is $3$, but $\text{lcm}(1,2,3)=6$. Then $A_5$ has exponent $30$, but $\text{lcm}(1,2,3,4,5)=60$. It seems to be right for even $n$, so one would think perhaps it's just off by a factor of $1/2$ for the odd $n$ - but then, it works for $A_7$! After that it goes back to being off again with $\text{Exp}(A_9)=1260$ whereas $\text{lcm}\{1,\ldots,9\}=2520$.

So, what is the formula for $\text{Exp}(A_n)$? How can we prove it?

Answer 1

The order of a permutation is the lcm of the sizes of the cycles appearing in its disjoint cycle factorization. The lcm of the alternating group is, then, the lcm of the sizes of the cycles of the disjoint cycle decompositions of its elements.

If $k\leq n$ is such that either $k$ is odd or $k+2\leq n$ then there is a permutation in $A_n$ which has a cycle of length $k$; indeed, in the first case, we can take it to be a cycle, in the second case a cycle times a transposition. On the other hand, if $k\leq n$ is even and $k+2>n$, then we must have either $k=n-1$ and $n$ odd or $k=n$ and $n$ even, and there are no elements in $A_n$ with a cycle of size $k$.

• Show that this gives the sizes of all possible cycles appearing in elements of $A_n$.

• Find a formula for the number you want.

Answer 2

If $n\geqslant2$, then the exponent of $A_n$ usually equals the exponent of $S_n$. Indeed, ${\rm exp}(A_n)={\rm exp}(S_n)$ if $n\not\in\{2^k,2^k+1\}$ for some $k\geqslant 1$, and ${\rm exp}(A_n)=\frac12{\rm exp}(S_n)$ otherwise. For a formula for ${\rm exp}(S_n)$ see: General formula to compute the exponent of the symmetric group $S_n$ .

Proof (sketch): ${\rm exp}(A_n)=\prod_{p\leqslant n}p^{k_p}$ where $k_p=\max\{k\mid \text{$A_n$ has a cycle of length $p^{k}$}\}$. A cycle of length $2^k\leqslant n$ lies in $A_n$ if and only if $2^k\leqslant n-2$, and one of odd length $p^k\leqslant n$ always lies in $A_n$.