Subgroups Cosets Lagrange Theorems

Let be a group and any subset of . Let be any two elements of . Now being a member... Click here to read more

The necessary and sufficient conditions for a subset of a group to be a subgroup are stated in the following... Click here to read more

Theorem 1: The intersection of two subgroups of a group is a subgroup of . Proof: Let and be any... Click here to read more

Theorem 1: Every subgroup of a cyclic group is cyclic. Proof: Let be a cyclic group generated by . Let... Click here to read more

If is a group, is a subgroup and is any element in , then the set is called the right... Click here to read more

Let be a subgroup of group . We know that no right coset of in is empty and any two... Click here to read more

Theorem 1: If , then the right (or left) coset or of is identical to , and conversely. Proof: Let... Click here to read more

Let be a subgroup of a group . If the element of belongs to the right coset , i.e. if... Click here to read more

Lagrange’s Theorem The order of a subgroup of a finite group divisor of the order of the group. Proof: Let... Click here to read more

Let us consider the set of all complexes of a group , which is nothing but a power set of... Click here to read more

Example 1: Find the proper subgroups of the multiplicative group of the sixth roots of unity. Solution: From trigonometry we... Click here to read more