# 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 […]

• ### Necessary and Sufficient Condition for a Subgroup

The necessary and sufficient conditions for a subset of a group to be a subgroup are stated in the following two theorems. Theorem 1: A subset of a group is a subgroup if and only if (i) and (ii) Proof: Suppose is a subgroup of then must be closed with respect to composition in , […]

• ### 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 […]

• ### Relation of Congruence Modulo a Subgroup in a Group

Let be a subgroup of a group . If the element of belong to the right coset , i.e. if , i.e., if then it is said that is congruent to modulo . Definition: Let be a subgroup of a group . For we say that is congruent to if and only if . Symbolically, […]

• ### 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 […]