Pullback of Quotient Group Isomorphism/Examples/Alternating Subgroups of Symmetric Groups

Example of Pullback of Quotient Group Isomorphism

Let $S_m$ and $S_n$ be symmetric groups on $m$ and $n$ letters respectively.

Let $A_m$ and $A_n$ be the alternating groups on $m$ and $n$ letters respectively.

Let $\theta: S_m / A_m \to S_n / A_n$ be an isomorphism.

The pullback of $S_m$ and $S_n$ by $\theta$ is a subset of $S_m \times S_n$ of the form:

$S_m \times^\theta S_n = \set {\tuple {\rho, \sigma}: \map \sgn \rho = \map \sgn \sigma}$

where $\map \sgn \rho$ denotes the sign of $\rho$.

Proof

From Alternating Group is Normal Subgroup of Symmetric Group, $A_m$ is normal in $S_m$.

Similarly with $A_n$.

From Order of Alternating Group it follows that:

$\index {S_m} {A_m} = 2$

and similarly for $A_n$.

Hence the result from Pullback of Quotient Group Isomorphism: Subgroups of Index 2.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $13$: Direct products: Example $13.12$