Subgroups Cosets Lagrange Theorems

  • Subgroups

    Let be a group and any subset of . Let be any two elements of . Now being members of the product of surely belongs to , but it may or may not belongs to . If, however, belongs to , we say that is stable for the composition in and the composition in has […]

  • Properties of Subgroups

    Theorem 1: The intersection of two subgroups of a group is a subgroup of . Proof: Let and be any two subgroups of . Then because at least the identity element is common in both and . Now to prove that is a subgroup of , it is sufficient to show that , , being […]

  • Subgroups of Cyclic Groups

    Theorem 1: Every subgroup of a cyclic group is cyclic. Proof: Let be a cyclic group generated by . Let be a subgroup of . Now every element of , hence also of , has the form , being an integer. Let be the smallest possible integer such that . We claim that . For […]

  • Cosets

    If is a group, is a subgroup and be any element in then the set is called the right coset generated by and is denoted by . Similarly, the set is called left coset generated by and is denoted by . Since , we see that itself is a right as well as left coset. […]

  • Coset Decomposition

    Let be a subgroup of group . We know that no right coset of in is empty and any two right cosets of in are either disjoint or identical. The union of all right cosets of in is equal to . Hence the set of all right cosets of in gives a partition of . […]

  • Properties of Cosets

    Theorem 1: If , then the right (or left) cosets or of is identical with , and conversely. Proof: Let be an arbitrary element of so that . Again, since is a subgroup, we have Thus every element of is also an element of . Hence Again This shows that every element of is also […]

  • Lagrange Theorem

    Lagrange’s Theorem: The order of a subgroup of a finite group divisor of the order of the group. Proof: Let be any subgroup of order of a finite group of order . Let us consider the coset decomposition of relative to . We will first show that each coset consists of different elements. Let , […]

  • Algebra of Complexes of a Group

    Let us consider the set of all complexes of a group , which in nothing but power set of . Let it be denoted by . Now, we define three binary composition in . The two compositions namely union and intersection of sets are familiar ones. How we define the multiplication of complexes. Multiplication is […]

  • Examples Subgroup of Cyclic Groups

    Example 1: Find the proper subgroups of the multiplicative group of the sixth roots of unity. Solution: From trigonometry we know that six sixth roots of unity are i.e. where If and be its proper subgroups, then and Example 2: Find all the subgroup of a cyclic group of order . Solution: We know that […]