Groups and Subgroups: A Formal Description A group is a non-empty set G together with an operation that combines any two elements a and b in G to form...
A group is a non-empty set G together with an operation * that combines any two elements a and b in G to form another element in G, satisfying specific properties. These properties include:
Closure: For any a, b, c in G, if a * b ∈ G, then also a * c ∈ G.
Associativity: For any a, b, c in G, we have (a * b) * c = a * (b * c).
Identity element: There exists an element e in G such that, for any a in G, we have a * e = a and e * a = a.
Groups are formally represented by ordered pairs (G, *) where G is the set and * is the operation.
A subgroup of a group G is a non-empty subset H of G that is closed under the operation *. In other words, if a and b are in H, then also a * b ∈ H.
For example, consider the group G = {1, 2, 3} under addition. The operation * is defined by a * b = min{a, b}.
The group G with addition is a group.
The subgroup H = {1} is a subgroup of G.
Subgroups play a significant role in studying the properties of groups. They can be used to test group membership of elements, solve group problems, and gain insights into the structure of groups