rank_by_country <- function(data,column,rank){
    ## We save the levels of column 2, the countries' names, in the countries vector
    countries <- levels(data[,2])
    ## We generate an empty vector that we will fill later, row by row, to generate our final output
    output <- vector()
    ## For loop to get the right data on each city. length(countries) is the number of different countries in our
    ## database. In our case we have 3 countries: China, UK, USA.
    for (i in 1:length(countries)) {
        ## countrydata subsets data by the considered country
        countrydata <- data [grep(countries[i],data$countries),]
        orderdata <- countrydata[order(decreasing = TRUE, countrydata[,column]),]
        ## append() adds elements at the end of a vector. We want to add the name of the city [rank,1],
        ## the areakm2 [rank,2] and the populationk [rank,3]. We don't add the name of the countries, because it
        ## will be the label of the rows.
        output <- append (output, as.character(orderdata[rank,1]))
        for (l in 3:4){
            output <- append (output, as.character(orderdata[rank,l]))
        }
    }
    ## Just because it's simpler to generate a matrix rather than a data frame, I generate it first and convert it
    ## to data frame immediatly after. 
    output <- as.data.frame(matrix(output,length(countries),3, byrow = TRUE))
    ## Name of the columns will be "cities", "areakm2" and "populationk". Name of the rows are the countries.
    colnames(output) <- c("cities","areakm2","populationk")
    rownames(output) <- countries
    return(output)
}

> rank_by_country(data,3,1)
        cities areakm2 populationk
China Shanghai    2643       21766
UK      London    1737        9789
USA         LA    1302        3884
> rank_by_country(data,3,2)
          cities areakm2 populationk
China    Beijing    1368       21500
UK    Manchester     116         255
USA          NYC    1214        8406