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The three-cycle subgroup of $Sym ({1, 2, 3})$ is normal and its quotient has two elements

Example

In $G = Sym ({1, 2, 3})$ , let $N = ⟨ (123) ⟩$ . Then

$N = {id, (123), (132)}$

is normal, and $G / N$ is the two-element group ${N, (12) N}$ .

Facts & Assumptions

Given: Permutations are composed from right to left.

[L1]

The symmetric group on ${1, 2, 3}$ is a group under composition (The symmetric group $Sym (X)$ : the bijections of a set $X$ under composition, $Sym (X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements).

[F1]

The subgroup generated by an element consists of its integral powers (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

[F2]

The index $[G : N]$ is the cardinality of the left-coset set (The coset set $G / H$ and the index $[G : H]$ of a subgroup).

[L2]

A subgroup of index two is normal (Every subgroup of index two is normal).

[L3]

A normal subgroup gives a quotient group under coset multiplication (For $N ⊴ G$ , the cosets form a group with identity $N$ and inverse $(g N)^{- 1} = g^{- 1} N$ ).

[L4]

A quotient with finite index has order equal to that index (If $[G : N]$ is finite then $∣ G / N ∣ = [G : N]$ ; for finite $G$ this equals $∣ G ∣ / ∣ N ∣$ ).

Verification

technique · direct

1.1

Since $(123)^{2} = (132)$ and $(123)^{3} = id$ , [F1] gives $N = {id, (123), (132)}$ .

L1F1algebra

2.1

The remaining three permutations are the transpositions, and $(12) N = {(12), (23), (13)}$ . Thus $N$ and $(12) N$ are the two left cosets of $N$ in $G$ .

step 1.1algebra

3.1

Consequently $[G : N] = 2$ by [F2], so $N ⊴ G$ by [L2].

step 2.1F2L2

4.1

The quotient group therefore exists by [L3], and its underlying coset set is precisely ${N, (12) N}$ . Its order is two by [L4].

step 2.1step 3.1L3L4∎

Depends on

Used by

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Sources

UCL lecture notes, Normal subgroups and quotients (standard reference, not scraped)

The three-cycle subgroup of Sym⁡({1,2,3})Sym({1,2,3}) is normal and its quotient has two elements