# Cayley table of a group

Just by looking at the cayley table of a group and looking at its subgroups, is their a theorem or something which tells you if the right and left cosets are equal?

I have question to do and I would love to half the workload by not having to to work out the same thing twice.
Thanks

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ok thanks

When multiplying cosets, can you just quite simply multiply them together?
Our teacher said something about their could be a potential problem?
What could the problem be?

jbunniii
Homework Helper
Gold Member
When multiplying cosets, can you just quite simply multiply them together?
Our teacher said something about their could be a potential problem?
What could the problem be?
You can always multiply cosets, but the result is not necessarily a coset. In other words, the set of right or left cosets is not generally a group. Thus you can form the product ##aHbH##, which is the set of all elements of the form ##ah_1bh_2##, but this does not generally equal ##abH##, nor can it generally be written in the form ##gH## at all.

However, if ##H## is a normal subgroup, then the left and right cosets are the same (##aH = Ha##), so we just call them cosets, and the set of cosets does form a group. In this case, the product ##aHbH## can be simplified as follows:
$$aHbH = a(Hb)H = a(bH)H = abHH = abH$$

You can always multiply cosets, but the result is not necessarily a coset. In other words, the set of right or left cosets is not generally a group. Thus you can form the product ##aHbH##, which is the set of all elements of the form ##ah_1bh_2##, but this does not generally equal ##abH##, nor can it generally be written in the form ##gH## at all.

However, if ##H## is a normal subgroup, then the left and right cosets are the same (##aH = Ha##), so we just call them cosets, and the set of cosets does form a group. In this case, the product ##aHbH## can be simplified as follows:
$$aHbH = a(Hb)H = a(bH)H = abHH = abH$$
Thank you for the good explanation.