Cosets of subgroups are a set of elements that are formed by multiplying a fixed element (or elements) of a subgroup by all possible elements of the original group. In other words, a coset is a subset of the original group that is formed by multiplying each element of the subgroup by all possible elements of the original group.
Cosets of subgroups are not groups themselves, but they can have similar properties to groups. Each coset is a subset of the original group, and it may have its own identity element, inverse elements, and closure under multiplication. However, a coset does not necessarily have all the properties of a group, such as associativity or a unique identity element.
No, a coset of a subgroup cannot be a group. While a coset may share some similarities with a group, it does not have all the necessary properties to be considered a group. For example, a coset may not be closed under multiplication or may not have an identity element.
Cosets of subgroups are important in group theory because they allow us to examine the structure of a group. By studying the cosets of a subgroup, we can better understand the relationships between different elements in a group. Cosets are also used in the proof of Lagrange's theorem, which states that the order of a subgroup must divide the order of the original group.
Yes, every element in a group can be represented by a coset of a subgroup. This is because every element in a group can be written as a product of an element in the subgroup and an element in the original group. Therefore, each element in the original group belongs to at least one coset of the subgroup.