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# Group Theory-S3 table Watch

1. This is the group table I have created for group. Now, I want to find out the regular representation for the group element or what is I also have to write every steps in detail.
2. 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
3. (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 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 , by constructing the left and right cosets for all , and checking to see that in all cases

or:

b) for each subgroup , check that for all .

Note that the notation says that *set* equals *set* - it doesn't require that for all . (I guess we could express that as saying that must be a permutation of )

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

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

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

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