# Normal Subgroups

• ### Normal Subgroups

Let be an abelian group, the composition in being denoted multiplicatively. Let be any subgroup of . If is an element of , then is a right coset of in and is a left coset of in . Also is abelian, therefore we must have . However, it is possible that is not abelian, yet […]

• ### Theorems of Normal Subgroups

Theorem 1: A subgroup of a group is normal if and only if . Proof: Let , then . Therefore is a normal subgroup of . Conversely, Let be a normal subgroup of . Then Also . Therefore we have From (i) and (ii) we conclude that Theorem 2: A subgroup of a group is […]

• ### Centre of a Group

Definition: The set of all those elements of a group which commute with every element of is called the centre of the group . Symbolically Theorem: The centre of a group is a normal subgroup of . Proof: We have . First we shall prove that is a subgroup of . Let , then and […]

• ### Quotient Groups

Definition: If is a group and is a normal subgroup of group , then the set of all cosets of in is a group with respect to multiplication of cosets. It is called the quotient group or factor group of by . The identity element of the quotient group by . Theorem: The set of […]

• ### Examples of Quotient Groups

Example 1: If be a normal subgroup of a finite group , then prove that Solution: number of distinct right (or left) cosets of in , as is the collection of all right (or left) cosets of in       by Lagrange’s Theorem Example 2: Show that every quotient group of a cyclic group is cyclic […]

• ### Conjugacy in a Group

Conjugate Element: If , then is said to be a conjugate of in if there exist an element such that . Symbolically, we shall write for this and shall refer to this relation as conjugacy. Then for some   Theorem: Conjugacy is an equivalence relation in a group. Proof: (i) Reflexivity: Let , then , […]