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The below written lines are taken from book title " Visual Group Theory" by Natham Carter

The most general way to deconstruct a large group into two factors is called taking quotients. It reveals not only direct products but also semi direct products. It gives us deep insight into groups structure.

Question : Why quotients in groups needed? I have heard about division in numbers but in groups how it make sense? 8/4 = 2, but $C_4 / C_2$ ? How researcher in mathematics comes up with the idea of quotient?

I have seen this question Why the term and the concept of quotient group?

  • Possible duplicate of this: https://math.stackexchange.com/questions/2565113/why-is-the-fact-that-a-quotient-group-is-a-group-relevant – wilkersmon Feb 21 '18 at 18:08
  • "My group is too large to understand, but I sure do understand this normal subgroup of significance. If I quotient it out I will get a smaller group--hopefully one more understandable--that still shows some remnants of the original." – Randall Feb 21 '18 at 18:08
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    +1 for "I have heard about division in numbers but in groups " –  Feb 21 '18 at 18:17
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    It's similar to the way that arithmetic modulo some prime gives insight into arithmetic over the natural numbers. For example, if I know that some equation has no solutions modulo $3,$ it can't have any solutions in integers. – saulspatz Feb 21 '18 at 18:36
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    The concept of quotient groups arises from the much more general idea of taking the quotient by an equivalence relation. In full generality, this just works on sets and ignores all structure. When we then put some structure on the set, we seek those equivalence relations which allow us to still have the same type of structure on the quotient (in a natural way that relates to the original structure). For groups, these equivalence relations are precisely determined by normal subgroups. – Tobias Kildetoft Feb 22 '18 at 06:55

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my opinion and self experience is that the quotient groups are famous and very important for group theorists since the have important behave and they built by normal subgroups which are very very important in group theory.somehow quotient groups play the role of fraction in numbers(that is if G\N is quotient group, we called G without considering N inside) but the whole story is about algebraic theorist like normal groups and normal subgroups so the quotient groups are important to them.and be careful u never can compare quotient with divisions but in some groups. for example u can study $\frac{z}{n{z}}$ which z is integer numbers.

pershina olad
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