```module Data.Group where

import Data.Monoid

-- |A 'Group' is a 'Monoid' plus a function, 'invert', such that:
--
-- @a \<> invert a == mempty@
--
-- @invert a \<> a == mempty@
class Monoid m => Group m where
invert :: m -> m

instance Group () where
invert () = ()

instance Num a => Group (Sum a) where
invert = Sum . negate . getSum
{-# INLINE invert #-}

instance Fractional a => Group (Product a) where
invert = Product . recip . getProduct
{-# INLINE invert #-}

instance Group a => Group (Dual a) where
invert = Dual . invert . getDual
{-# INLINE invert #-}

instance Group b => Group (a -> b) where
invert f = invert . f

instance (Group a, Group b) => Group (a, b) where
invert (a, b) = (invert a, invert b)

instance (Group a, Group b, Group c) => Group (a, b, c) where
invert (a, b, c) = (invert a, invert b, invert c)

instance (Group a, Group b, Group c, Group d) => Group (a, b, c, d) where
invert (a, b, c, d) = (invert a, invert b, invert c, invert d)

instance (Group a, Group b, Group c, Group d, Group e) => Group (a, b, c, d, e) where
invert (a, b, c, d, e) = (invert a, invert b, invert c, invert d, invert e)

-- |An 'Abelian' group is a 'Group' that follows the rule:
--
-- @a \<> b == b \<> a@
class Group g => Abelian g

instance Abelian ()

instance Num a => Abelian (Sum a)

instance Fractional a => Abelian (Product a)

instance Abelian a => Abelian (Dual a)

instance Abelian b => Abelian (a -> b)

instance (Abelian a, Abelian b) => Abelian (a, b)

instance (Abelian a, Abelian b, Abelian c) => Abelian (a, b, c)

instance (Abelian a, Abelian b, Abelian c, Abelian d) => Abelian (a, b, c, d)

instance (Abelian a, Abelian b, Abelian c, Abelian d, Abelian e) => Abelian (a, b, c, d, e)
```