When learning about groups, it’s helpful to look at group multiplication tables. Sometimes called Cayley Tables, these tell you everything you need to know to analyze and work with small groups. It’s even possible to use these tables to systematically find all groups of small order!
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Abstract Algebra deals with groups, rings, fields, and modules. These are abstract structures which appear in many different branches of mathematics, including geometry, number theory, topology, and more. They even appear in scientific topics such as quantum mechanics.