lanedance said:trivial, but what if A is a subgroup of B?
lanedance said:yeah i suppose its really obvious, and the idea you give covers both 2 & 3
A group is a mathematical structure consisting of a set of elements and an operation that combines any two elements to form a third element, satisfying the group axioms of closure, associativity, identity, and inverse.
A subgroup is a subset of a group that also satisfies the group axioms. This means that the subgroup is closed under the group operation, contains the identity element, and every element in the subgroup has an inverse within the subgroup.
A group can be considered a union of subgroups if and only if all the elements in the group are contained within the subgroups and the subgroups together satisfy the group axioms. This is a feasible concept, but not all groups can be expressed as a union of subgroups.
To determine if a group is a union of subgroups, we can look at the structure of the group and identify any subgroups that exist within it. Then, we can check if the subgroups together satisfy the group axioms. If they do, then the group can be considered a union of subgroups.
Expressing a group as a union of subgroups can provide a clearer understanding of the structure of the group and can make certain operations and proofs easier. It can also help in identifying important subgroups within the group and their properties.