(i) Group: In mathematics, a group is a set equipped with a binary operation that combines any two elements of the set and satisfies certain properties. Formally, a group is defined as a tuple (G, โˆ—), where G is a set and โˆ— is a binary operation on G. The binary operation โˆ— must satisfy the following conditions:

  1. Closure: For any elements a and b in G, the result of the operation a โˆ— b is also in G.
  2. Associativity: For any elements a, b, and c in G, the operation is associative, meaning that (a โˆ— b) โˆ— c = a โˆ— (b โˆ— c).
  3. Identity element: There exists an identity element e in G such that for any element a in G, a โˆ— e = e โˆ— a = a.
  4. Inverse element: For every element a in G, there exists an inverse element aโปยน in G such that a โˆ— aโปยน = aโปยน โˆ— a = e.

These properties make a group a fundamental algebraic structure, capturing notions of symmetry, transformation, and symmetry breaking in various mathematical and scientific contexts.

(ii) Cyclic group: A cyclic group is a specific type of group that is generated by a single element. More formally, a group G is said to be cyclic if there exists an element g in G such that every element of G can be obtained by repeatedly applying the group operation to g and its powers. In other words, every element of the cyclic group is a power of the generator element g.

If a cyclic group is finite, it is denoted as โŸจgโŸฉ, where โŸจgโŸฉ represents the subgroup generated by g. The order of the cyclic group is equal to the number of distinct elements it contains, which is also the smallest positive integer n such that gโฟ = e, where e is the identity element of the group.

Cyclic groups have important applications in various branches of mathematics, such as number theory, abstract algebra, and cryptography. They provide a simple and structured framework for understanding and analyzing certain mathematical phenomena.