group
A group is an algebraic structure that contains a set and a binary operation defined on the set and satisfies the following four main properties.
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Closedness: For any two elements a and b in the group, their combination a * b must also belong to the group through the binary operation of the group. In other words, the result of the operation does not cause the element to leave the group.
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Associativity: The binary operations in the group are associative, that is, for any elements a, b and c, (a * b) * c = a * (b * c).
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Identity element: There is a specific element e in the group, called the identity element, which satisfies that for any element a in the group, e * a = a * e = a .
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reverse element: each element a in the group of groups, necessarily exist one inverse element a-1，Use a * a-1 = a-1 * a = e, of which e is the lowest element.
The order of the group: The number of elements in the group is called the order of the group G, denoted as |G|
The order (sometimes called the period) of element a in the group: the smallest positive integer k such that ak = e holds is the element a The order of (where e is the unit element of this group). If k does not exist, the order of a is said to be infinite. All elements of a finite group have finite order (Proof).
cyclic group
A cyclic group is a special group whose elements are generated by the repeated action of a generator. Specifically, if there is an element g in the group G that can generate all the elements in G by repeatedly using binary operations (usually multiplication or addition), then the group G is a cyclic group, and the element g is called a generator.
Formally, let (G,·) be a group. If there is an element g ∈ G, such that G = < g > = { gk | k ∈ Z }, then (G,·) forms a cyclic group. The group generated by any element in the group G is a cyclic group and a subgroup of G.
For example: modulo 8 additive group, {0, 1, 2, 3, 4, 5, 6, 7}, where 0 is the identity element and the generators are 1, 3, 5, 7
exchange group
A commutative group, also known as a commutative group or an Abelian group, is a group that satisfies the commutative property. The commutative property means that any two elements in the group can exchange places under the group operation, that is, for all elements a and b, a * b = b * a, where * represents the group operation.
Classic examples include the additive group (Z, +) on the set of integers, the additive group (R, +) on the set of real numbers, and the multiplicative group (Z, *) on the set of integers where * represents multiplication. These groups are all commutative groups because their operations satisfy the commutative property.