# How Do You List Elements of G/H in Z10 When H={α,β,δ}?

• MHB
• simo1
In summary, In summary, The subgroup of $S_3$ with order 3 is generated by the 3-cycle $(1\ 2\ 3)$, and its two cosets are $H$ and $Ha$ where $a$ is any permutation not in $H$. The 2-cycle (transposition) $(1\ 2)$ can be used to list the elements of one of the cosets, which are $(1\ 2), (1\ 3), (2\ 3)$.
simo1
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}

Re: clarity on groups

$\Bbb Z_{10}$ has no subgroup of 3 elements, since 3 does not divide 10. Did you have in mind some specific subgroup of $\Bbb Z_{10}$?

Re: clarity on groups

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 $\Bbb Z_{10}$?

my apologies I meant G=S3

Re: clarity on groups

Well, fortunately, there is only ONE subgroup of $S_3$ of order 3, which is generated by any 3-cycle.

The USUAL notation for such a 3-cycle is:

$(1\ 2\ 3)$ which is shorthand for the mapping:

$1 \to 2$
$2 \to 3$
$3 \to 1$.

The subgroup generated by this is:

$H = \{e, (1\ 2\ 3), (1\ 3\ 2)\}$.

Lagrange's Theorem tells us there will be exactly TWO cosets, $H$ and $Ha$ where $a$ is any permutation not in $H$. The 2-cycle (transposition) $(1\ 2)$ will do, and we find:

$H(1\ 2) = \{(1\ 2), (1\ 2\ 3)(1\ 2), (1\ 3\ 2)(1\ 2)\}$

$= \{(1\ 2), (1\ 3), (2\ 3)\}$

I would explain that G/H represents the quotient group, where the elements of H are the cosets of G. In this case, H is a subgroup of G, meaning that it is a subset of G that also forms a group under the same operation. The elements of G/H are the distinct cosets of H in G, which are represented by the elements α, β, and δ. These elements are not individual elements of G, but rather represent the sets of elements in G that are related by the operation defined by H. Therefore, the elements of G/H are not listed in the same way as the elements of G, but rather as cosets of H.

## 1. What are the "Elements of G/H in Z10: α, β, δ"?

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.

## 2. How are α, β, and δ related to the quotient group G/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.

## 3. Can α, β, and δ be elements of both G and 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.

## 4. How do α, β, and δ affect the structure of G/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.

## 5. Are there any special properties of the elements α, β, and δ in G/H?

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.

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