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

Example

In $G=\operatorname{Sym}(\{1,2,3\})$ , let $N=\langle(123)\rangle$ . Then

N=\{\operatorname{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 $\operatorname{Sym}(X)$ : the bijections of a set $X$ under composition, $\operatorname{Sym}(X)$ 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 $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$ , 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 $N\mathrel{\trianglelefteq}G$ , the cosets form a group with identity $N$ and inverse $(gN)^{-1}=g^{-1}N$ ).

[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 $(123)^3=\operatorname{id}$ , [F1] gives $N=\{\operatorname{id},(123),(132)\}$ .

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 $N\trianglelefteq G$ 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})\operatorname{Sym}(\{1,2,3\})Sym({1,2,3}) is normal and its quotient has two elements