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Normal Sub-Group

A sub-group

Now,

Alternatively, a subgroup

It is important to note that for a group

and

Multiplying by g on right and g-1 on left of (1), we get

From

**N**of**G**is said to be a normal sub-group of**G**if for every**g G**and**n N**,**gng**. The normal subgroup^{-1}N**N**of**G**is denoted by**N Δ G**.Now,

**gNg**, for some^{-1}= {gng-1, n N)**g G**.Alternatively, a subgroup

**N**is said to be a normal sub-group of**G**if and only if**gNg**for every^{-1}⊂ N**g G**.It is important to note that for a group

**G**, the sub-groups of**G**and**(e)**are always normal. These subgroups are called trivial or improper normal subgroup.**A group**

Simple group:Simple group:

**G ≠ (e)**is said to be simple group if it has no proper normal sub-groups.**Theorem: N**is a normal subgroup of**G**if and only if**gNg**for every^{-1}= N**g G**.**Proof:**Now it is given that**gNg**for every^{-1}= N**g G**. Also**gNg-1 = N****gNg-1 ⊂ N****N**is a normal sub-group in**G**.**Conversely: N**is a normal subgroup in**G gNg**^{-1}⊂ Nand

**g**^{-1}Ng = g^{-1}N (g^{-1})^{-1}⊂ N (1)Multiplying by g on right and g-1 on left of (1), we get

**gg**^{-1}N (g^{-1})^{-1}g^{-1}⊂ gNg^{-1}eNe⊂gNg^{-1}**N⊂gNg**^{-1}(2)From

**(1)**and**(2)**, we get**gNg**.^{-1}= N**Normal Sub-Group Homework | Normal Sub-Group Homework Help | Normal Sub-Group Homework Help Services | Live Normal Sub-Group Homework Help | Normal Sub-Group Homework Tutors | Online Normal Sub-Group Homework Help | Normal Sub-Group Tutors | Online Normal Sub-Group Tutors | Normal Sub-Group Homework Services | Normal Sub-Group**

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