  Let $A_n\subseteq S_n$ denote the alternating and the symmetric groups on
$1,...,n$. MacMahaon's theorem, about the equi-distribution of the length and
the major indices in $S_n$, has received far reaching refinements and
generalizations, by Foata, Carlitz, Foata-Schutzenberger, Garsia-Gessel and
followers. Our main goal is to find analogous statistics and identities for the
alternating group $A_{n}$. A new statistic for $S_n$, {\it the delent number},
is introduced. This new statistic is involved with new $S_n$ equi-distribution
identities, refining some of the results of Foata-Schutzenberger and
Garsia-Gessel. By a certain covering map $f:A_{n+1}\to S_n$, such $S_n$
identities are `lifted' to $A_{n+1}$, yielding the corresponding $A_{n+1}$
equi-distribution identities.
