47 lines
1.4 KiB
C#
47 lines
1.4 KiB
C#
namespace Skeleton.Algebra.Groups;
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/// <summary>
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/// a set of elements closed under group operation
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/// </summary>
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/// <typeparam name="TElement"></typeparam>
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public abstract class Group<TElement>
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{
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/// <summary>
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/// unit of the group<br/>
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/// unit * x = x * unit = x for all x in group
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/// </summary>
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public abstract TElement GroupUnit { get; }
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/// <summary>
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/// inverse of the element under group operation
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/// </summary>
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/// <param name="element"></param>
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/// <returns></returns>
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public abstract TElement GroupInv(TElement element);
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/// <summary>
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///
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/// </summary>
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/// <param name="a"></param>
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/// <param name="b"></param>
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/// <returns></returns>
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public abstract TElement GroupOperation(TElement a, TElement b);
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/// <summary>
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/// Commutator of group
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/// </summary>
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/// <param name="a">a</param>
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/// <param name="b">b</param>
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/// <returns></returns>
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public TElement GroupCommutator(TElement a, TElement b) =>
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GroupOperation(GroupOperation(a, b), GroupOperation(GroupInv(a), GroupInv(b)));
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/// <summary>
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/// a conjugate of b => aba^{-1}
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/// </summary>
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/// <param name="a">a</param>
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/// <param name="b">b</param>
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/// <returns>a b a^{-1}</returns>
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public TElement Conjugation(TElement a, TElement b) =>
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GroupOperation(a, GroupOperation(b, GroupInv(a)));
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}
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