Group Theory (by Abstract Algebra)  Exercise 14 solution pdf

   Exercise 14

        👉Normal Subgroup

        👉 Quotient group

Normal subgroup:

    We absorve that when G=(Z,+) and H=(3Z,+) and, H=(3Z,+), each left coset of H  is also a right coset of H ;

when G=S3 and H={po,p1,p2), each left coset of H is also a right coset of H . But when G=S3 and H={po,p3}, H

is a left coset as well as a right coset 

and other left cosets are not right cosets .

      Thus we see that for some subgroup the left cosets and and the right cosets  and right cosets differ.

   

Defination.Normal subgroup:

       A subgroup H of a group G is said to be a normal subgroup of G if Ha=aH holds for all a in G.

The standard notation for "H is a normal subgroup of G ,is H∆G.

Note 1. 

The condition aH = Ha does not demand that for every h €H,

   ah=ha.

Note 2.

When H is a normal subgroup of a group G, there is no distinction between the left cosets and the right cosets of H and we speak simply of "the cosets of H". 

 Note 3. 

The improper subgroup G of a group G is a normal subgroup of G.

Proof. 

Let a € G. Then aG = G and Ga = G. Therefore aG= Ga holds for all a € G. This proves that G is normal in G.

Note 4.

 The trivial subgroup of a group G is a normal subgroup of G. 

Proof ,

Let H = {e}, the trivial subgroup of G and let a € G. Then aH={a}and Ha ={a}. Therefore aH = Ha and this holds for all a€G This proves that H is normal in G.

definition. Simple group

A group G is said to be a simple group if G has no normal subgroups other than the trivial and the improper subgroups of G. For example, a group of prime order is a simple group, since the only subgroups (and therefore the only normal subgroups) of such a group are the trivial subgroup and the improper subgroup.

Quotient group.

Let H be a normal subgroup of a group G. Since H is normal we need not make any distinction between the left cosets and the right cosets. La S be the set of all distinct cosets of H in G. 

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Group theory এর 6 নম্বর চ্যাপ্টার থেকে 14 নম্বর পর্যন্ত দিয়ে দেয়ছি।। 

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