if you consider how the composition table of a group is developed, each element in the set is mapped to every element of the set, thus each row and each column contains every element in the set forming a group under composition. The sub group has the special property of being the sub set containing the identity element e. The mode of construction of the group guarantees that the index of the subgroup will be a multiple of the index of the entire group as each subgroup and subset corresponds to a column or a row within the composition table, each of which has the same number of elements as the set.
Could also prove the claim using Lagrange’s theorem and the fact that there are inherent divisibilities of magnitudes of the group and the given subgroups (number theory).
Tetro I’n this example that’s incorrect, H is a subgroup, not a coset, then he picks an element a outside of H to show that the element times the subgroup creates the main group G. If a was inside of H you would not be able to generate the main group G, hence H would not be a coset, in this example, bc it requires multiple partitions for aH to create G.
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if you consider how the composition table of a group is developed, each element in the set is mapped to every element of the set, thus each row and each column contains every element in the set forming a group under composition. The sub group has the special property of being the sub set containing the identity element e. The mode of construction of the group guarantees that the index of the subgroup will be a multiple of the index of the entire group as each subgroup and subset corresponds to a column or a row within the composition table, each of which has the same number of elements as the set.
Could also prove the claim using Lagrange’s theorem and the fact that there are inherent divisibilities of magnitudes of the group and the given subgroups (number theory).
You never say whether a is an element of H or not. Not obvious to someone seeing cosets for the first time.
a is outside of H so it’s not an element of H. It would’nt be a co set if a was an element of H
@@AstroKidLo actually, H is a coset, just a special one since it contains e, so if a was in H, it would still be a coset
Tetro I’n this example that’s incorrect, H is a subgroup, not a coset, then he picks an element a outside of H to show that the element times the subgroup creates the main group G. If a was inside of H you would not be able to generate the main group G, hence H would not be a coset, in this example, bc it requires multiple partitions for aH to create G.
@@AstroKidLo you're right, I thought about a more general case
neat