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Chapter 15: Groups and Symmetries

      https://doi.org/10.1142/9789813275386_0015Cited by:0 (Source: Crossref)
      Abstract:

      A group G is a set of objects { a, b, c, … } (not necessarily countable) together with a binary operation which associates with any ordered pair of elements a, b in G a third element ab in G (closure). The binary operation (called group multiplication) is subject to the following requirements:

      • There exists an element e in G called the identity element such that eg = ge = g for all gG.

      • For every gG there exists an inverse element g−1 in G such that gg−1 = g−1g = e.

      • Associative law. The identity (ab)c = a(bc) is satisfied for all a, b, cG.

      If ab = ba for all a, bG we call the group commutative. If G has a finite number of elements it has finite order n(G), where n(G) is the number of elements. Otherwise, G has infinite order…