namespace Skeleton.Algebra.Groups;

/// <summary>
/// a set of elements closed under group operation
/// </summary>
/// <typeparam name="TElement"></typeparam>
public abstract class Group<TElement>
{
    /// <summary>
    /// unit of the group<br/>
    /// unit * x = x * unit = x for all x in group
    /// </summary>
    public abstract TElement GroupUnit { get; }
    /// <summary>
    /// inverse of the element under group operation
    /// </summary>
    /// <param name="element"></param>
    /// <returns></returns>
    public abstract TElement GroupInv(TElement element);
    /// <summary>
    /// 
    /// </summary>
    /// <param name="a"></param>
    /// <param name="b"></param>
    /// <returns></returns>
    public abstract TElement GroupOperation(TElement a, TElement b);

    /// <summary>
    /// Commutator of group
    /// </summary>
    /// <param name="a">a</param>
    /// <param name="b">b</param>
    /// <returns></returns>
    public TElement GroupCommutator(TElement a, TElement b) =>
        GroupOperation(GroupOperation(a, b), GroupOperation(GroupInv(a), GroupInv(b)));
    /// <summary>
    /// a conjugate of b => aba^{-1}
    /// </summary>
    /// <param name="a">a</param>
    /// <param name="b">b</param>
    /// <returns>a b a^{-1}</returns>
    public TElement Conjugation(TElement a, TElement b) =>
        GroupOperation(a, GroupOperation(b, GroupInv(a)));
    

}