Thanks to all.
I summarized (in order to thank the list) the solutions to help future workers searching subjects like this at R help.
 
# Number of rows
nr = 10
# Data set
dataf = as.data.frame(matrix(c(rnorm(nr),rnorm(nr)*2,runif(nr),sort(c(1,1,2,2,3,3,sample(1:3,nr-6,replace=TRUE)))),ncol=4))
names(dataf)[4] = "class"
#-----------------------------------------------------
#Solution 1:
#works, but need space to allocate the new data: splidata
 
# Splitting your data
splitdata = split(dataf,dataf$class)
# Correlations
correl=lapply(splitdata,function(x) cor.test(x[,1],x[,2])$estimate)
res=do.call(c,correl)
names(res)=paste('class',unique(dataf[,4]),sep="")
res

#-----------------------------------------------------
#Solution 2:
sapply(by(dataf[,c("V1","V2")], dataf$class, cor), '[', 3)
 
#works, and I can specify parameters like
sapply(by(dataf[,c("V1","V2")], dataf$class, cor, method="pearson"), '[', 3)

#-----------------------------------------------------
#Solution(s) 3:
tapply(rownames(dataf), dataf$class, function(r) cor(dataf[r, "V1"],dataf[r, "V2"]))
#or tapply(rownames(dataf), dataf$class, function(r) with(dataf[r, ], cor(V1, V2)))
 
#works good; I can directly specify parameters at "function()" like
tapply(rownames(dataf), dataf$class, function(r) with(dataf[r, ], cor(V1, V2, method="kendall")))
 
#but, in a 100 thousand rows data, it is TEN times slower than sapply (I surprised)
#note that tapply(rownames(dataf), dataf$class, function(r) with(dataf[r, ], cor(V1, V2))) is SIX times slower than sapply

#-----------------------------------------------------
#Solution(s) 4:
install.packages("plyr")
library(plyr)
 
ddply(dataf, .(class), function(df) data.frame(cor(df[, 1:2])))
daply(dataf, .(class), function(df) cor(df[, 1:2]))
dlply(dataf, .(class), function(df) cor(df[, 1:2]))
 
#Interesting library, but not good to my example - I need a vector with results 
#[only cor(V1,V2) and not cor(V1,V1)], that is, I don't want rectangular objects



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