Conditionally determining value of column by looking at last group

We can nest after grouping by 'Group', then remove the first and last elements of the 'data' column, use map2 to do a comparison of corresponding elements and then append with the FALSE elements for the first group

library(dplyr)
library(purrr)
df2 <- df1 %>%
         group_by(Group) %>%
         nest

flag <-  map2(df2$data[-1], df2$data[-nrow(df2)], ~ 
      .x$Value %in% .y$Value) %>%
      unlist
df1$Last_Group <- c(rep(FALSE, nrow(df2$data[[1]])), flag)

You can use a join to lookup values in the previous group to see if those values exists. It should be faster than looping through the groups. I am not familiar with tidyverse but here is an implementation in data.table (which should also be faster than tidyverse if your data is large enough):

library(data.table)
setDT(DF)
DF[, c("g", "pg") := .(r <- rleid(Group), r - 1L)]
DF[, ilg := FALSE][DF, on=.(pg=g, Value), ilg := TRUE]

output (note that there is a typo for Value in row 12 of OP's desired output):

    Group Value g pg   ilg
 1:     a     1 1  0 FALSE
 2:     a     2 1  0 FALSE
 3:     a     3 1  0 FALSE
 4:     a     4 1  0 FALSE
 5:     b     5 2  1 FALSE
 6:     b     2 2  1  TRUE
 7:     b     3 2  1  TRUE
 8:     c     6 3  2 FALSE
 9:     c     7 3  2 FALSE
10:     c     8 3  2 FALSE
11:     c     3 3  2  TRUE
12:     c     6 3  2 FALSE
13:     d     9 4  3 FALSE
14:     d    10 4  3 FALSE
15:     e     9 5  4  TRUE

data:

DF <- structure(list(Group = structure(c(1L, 1L, 1L, 1L, 2L, 2L, 2L, 
3L, 3L, 3L, 3L, 3L, 4L, 4L, 5L), .Label = c("a", "b", "c", "d", 
"e"), class = "factor"), Value = c(1, 2, 3, 4, 5, 2, 3, 6, 7, 
8, 3, 6, 9, 10, 9)), .Names = c("Group", "Value"), row.names = c(NA, 
-15L), class = "data.frame")