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simo1
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someone had a post on finite quotient groups. i understood that but how does one list elements of G/H if H is a subgroup of G.
where:
G=Z10 H={α,β,δ}
where:
G=Z10 H={α,β,δ}
Deveno said:$\Bbb Z_{10}$ has no subgroup of 3 elements, since 3 does not divide 10. Did you have in mind some specific subgroup of $\Bbb Z_{10}$?
The elements of G/H in Z10 refer to the elements of the quotient group G/H, where G is a group and H is a subgroup, in the ring of integers modulo 10. The elements α, β, and δ represent the cosets of H in G, which are the distinct subsets of G obtained by partitioning G into equivalence classes based on the operation of H.
The elements α, β, and δ are representatives of the cosets of H in G. This means that they represent the entire coset and any element within that coset can be obtained by adding or multiplying by elements of H. In other words, α, β, and δ are the building blocks of the quotient group G/H.
Yes, α, β, and δ can be elements of both G and H. In fact, they are always elements of G because they represent the cosets of H in G. However, they may also be elements of H if they are the identity element or if they are contained within a subgroup of H.
The elements α, β, and δ play a crucial role in determining the structure of G/H. They help to identify the distinct cosets of H in G and how they relate to each other. The operations of G/H are defined based on the operations of G and H, which are reflected in the elements α, β, and δ.
Yes, there are several special properties of these elements in G/H. For example, they can be used to define the order and index of the subgroup H, as well as the order of the quotient group G/H. They also play a role in determining the coset representatives and the Lagrange's theorem for subgroups.