Find cyclic subgroups of finite order of groups with integer elements under modular multiplication

Main.hs

{-# LANGUAGE RecordWildCards #-}
module Main where

data Solution = Solution
  { remainder :: Integer
  , element :: Integer
  , power :: Integer
  }

instance Show Solution where
  show Solution{..} =  (show remainder)
                    ++ " = "
                    ++ (show element)
                    ++ "^{" ++ (show power) ++ "}"

cyclic :: (Integer -> Integer -> Integer) -- Group operation
       -> Integer -- Identity of the group
       -> Integer -- Element of group
       -> [Solution]
cyclic op i e = map (uncurry (flip Solution e)) values
  where
    pows = iterate (\(rem, y) -> ((rem `op` e), y + 1)) (e, 1)
    values = takeWhileInclusive ((i /=) . fst) pows

takeWhileInclusive :: (a -> Bool) -- Predicate
                   -> [a] -- Input
                   -> [a] -- Output
takeWhileInclusive p = foldr (\ x acc -> x : if p x then acc else []) []

main :: IO ()
main = do
  putStrLn "hello world"

Updated so it takes an arbitrary function for the group.

In the first version, for U(14) you'd do

*Main> cyclic 3 14
[3 = 3^{1},9 = 3^{2},13 = 3^{3},11 = 3^{4},5 = 3^{5},1 = 3^{6}]

Now you do

*Main> cyclic (\ x y -> (x*y) `mod` 14) 1 3
[3 = 3^{1},9 = 3^{2},13 = 3^{3},11 = 3^{4},5 = 3^{5},1 = 3^{6}]