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- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/02%3A_Groups/2.03%3A_The_Definition_of_a_GroupIn summary, we used associativity, identity elements, and inverses in a set of all integers to solve the given equation. This perhaps suggests that these would be useful traits for a binary structure ...In summary, we used associativity, identity elements, and inverses in a set of all integers to solve the given equation. This perhaps suggests that these would be useful traits for a binary structure and/or its operation to have. They are in fact so useful that a binary structure displaying these characteristics is given a special name. We note that these axioms are rather strong; “most” binary structures aren't groups.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/02%3A_Groups/2.01%3A_Examples_of_groupsGroups are one of the most basic algebraic objects, yet have structure rich enough to be widely useful in all branches of mathematics and its applications. A group is a set G with a binary operati...Groups are one of the most basic algebraic objects, yet have structure rich enough to be widely useful in all branches of mathematics and its applications. A group is a set G with a binary operation G×G→G that has a short list of specific properties. Before we give the complete definition of a group in the next section, this section introduces examples of some important and useful groups.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/02%3A_Introduction_to_Groups/2.03%3A_GroupsIn groups, it turns out that inverses are always “two-sided". That is, if G is a group and g,h∈G such that gh=e, then it must be the case that hg=e, as well. Notice all that we have...In groups, it turns out that inverses are always “two-sided". That is, if G is a group and g,h∈G such that gh=e, then it must be the case that hg=e, as well. Notice all that we have done is taken the statements of Definition: Exponents, which use multiplicative notation for the group operation, and translated what they say in the case that the group operation uses additive notation.
