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    $\begin{matrix}

 &  e&  a&  b&  c&  d&f \\ 

 e&  e& a &  b&  c&  d&f \\ 

 a&  a&  b&  e&  d&  f&c \\ 

 b&  b&  e&  a&  f&  c&d \\ 

 c&  c&  f&  d&  e&  b&a \\ 

 d&  d&  c&  f&  a&  e&b \\ 

 f&  f&  d&  c&  b&  a&e 

\end{matrix}$

    This is the group table I have created for S_3 group. Now, I want to find out the regular representation for the group element c or what is D_{ij} I also have to write every steps in detail.
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    I worked out the answer to the first question, but can anyone tell me how to find out the normal sub-group and the factor group from the above group table
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    (Original post by roshanhero)
    I worked out the answer to the first question, but can anyone tell me how to find out the normal sub-group and the factor group from the above group table
    I don't think that there's any magical easy way so:

    1. Find all of the sub-groups of S_3 either by staring at the group table till you find them, or by cheating and googling "sub-groups of s3".

    2. Check each of them for normality either:

    a) for each sub-group H, by constructing the left and right cosets aH, Ha for all a \in S_3, and checking to see that aH=Ha in all cases

    or:

    b) for each subgroup H, check that aHa^{-1} = H for all a \in S_3.

    Note that the notation aH=Ha says that *set* aH equals *set* Ha - it doesn't require that ah=ha for all h \in H. (I guess we could express that as saying that Ha must be a permutation of aH)

    As for the factor group, I think that you're done once you've found a normal subgroup H, since it's guaranteed that the cosets of that group form the factor (quotient) group named S_3 / H under the operation of coset multiplication.

    You may find this article useful:

    http://gowers.wordpress.com/2011/11/...otient-groups/

    (Edit: actually, there's another very useful fact in this case: a subgroup H is normal if its index G\text{:}H (i.e. the number of cosets of H) is 2. And what does Lagrange's theorem tell you about what order H must have then?)
 
 
 
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