Definition:Pullback of Quotient Group Isomorphism

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Definition

Let $G, H$ be groups.

Let $N \lhd G, K \lhd H$ be normal subgroups of $G$ and $H$ respectively.

Let:

$G / N \cong H / K$

where:

$G / N$ denotes the quotient of $G$ by $N$
$\cong$ denotes group isomorphism.

Let $\theta: G / N \to H / K$ be such a group isomorphism.


The pullback $G \times^\theta H$ of $G$ and $H$ via $\theta$ is the subset of $G \times H$ of elements of the form $\tuple {g, h}$ where $\map \theta {g N} = h K$.


Examples

Quotient Groups of Order 2 Subgroups

Let $G$ and $H$ be groups.

Let $N$ and $K$ be normal subgroups of $G$ and $H$ respectively such that:

their quotient groups $G / N$ and $H / K$ are isomorphic
their indices are $2$:
$\index G N = \index H K = 2$

Let $\theta: G / N \to H / K$ be an isomorphism.


The pullback of $G$ and $H$ by $\theta$ is a subset of $G \times H$ of the form:

$G \times^\theta H = \set {\tuple {g, h}: \paren {g \in N, h \in K} \text { or } \paren {g \notin N, h \notin K} }$


Alternating Subgroups of Symmetric Groups

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 $\rho$.


Cyclic Group $C_6$ with Alternating Group $A_4$

Let $G = C_6$ be the cyclic group of order $6$:

$G = \gen x$

Let $H = A_4$ be the alternating group on $4$ letters.

Let $e_G$ and $e_H$ denote the identity elements of $G$ and $H$ respectively.


Let $N$ be the subgroup of $G$:

$N = \gen {x^3}$

which from Subgroup of Abelian Group is Normal is normal.

Let $K$ be the subgroup of $H$:

$\set {e_H, \tuple {1 2} \tuple {3 4}, \tuple {1 3} \tuple {2 4}, \tuple {1 4} \tuple {2 3} }$

which from Klein Four-Group is Normal in A4 is normal.


Let $\theta: G / N \to H / K$ be the mapping defined as:

$\forall M \in G / N: \map \theta M = \begin{cases} K & : M = N \\ \tuple {1 2 3} K & : M = x N \\ \tuple {1 3 2} K & : M = x^2 N \end{cases}$


The pullback of $G$ and $H$ by $\theta$ is:

$G \times^\theta H = \set {\tuple {e_G, e_H}, \tuple {e_G, \tuple {1 2} \tuple {3 4} }, \tuple {e_G, \tuple {1 3} \tuple {2 4} }, \tuple {e_G, \tuple {1 4} \tuple {2 3} }, \\ \tuple {x^3, e_H}, \tuple {x^3, \tuple {1 2} \tuple {3 4} }, \tuple {x^3, \tuple {1 3} \tuple {2 4} }, \tuple {x^3, \tuple {1 4} \tuple {2 3} }, \\ \tuple {x, \tuple {1 2 3} }, \tuple {x, \tuple {1 3 4} }, \tuple {x, \tuple {2 4 3} }, \tuple {x, \tuple {1 4 2} }, \\ \tuple {x^4, \tuple {1 2 3} }, \tuple {x^4, \tuple {1 3 4} }, \tuple {x^4, \tuple {2 4 3} }, \tuple {x^4, \tuple {1 4 2} }, \\ \tuple {x^2, \tuple {1 3 2} }, \tuple {x^2, \tuple {1 4 3} }, \tuple {x^2, \tuple {2 3 4} }, \tuple {x^2, \tuple {1 2 4} }, \\ \tuple {x^5, \tuple {1 3 2} }, \tuple {x^5, \tuple {1 4 3} }, \tuple {x^5, \tuple {2 3 4} }, \tuple {x^5, \tuple {1 2 4} } }$


Also see


Sources