Subquotient

There are subquotients of groups which are neither a subgroup nor a quotient of it. For example, according to the article Sporadic group, Fi₂₂ has a double cover which is a subgroup of Fi₂₃, so it is a subquotient of Fi₂₃ without being a subgroup or quotient of it.

The relation subquotient of is an order relation, which shall be denoted by $\backslash preceq$. It shall be proved for groups.

;Notation

:Let {{Mvar|G}} be a group, let {{Math|G′}} be a subgroup of {{Mvar|G}}, let {{Math|G′′}} be a normal subgroup of {{Math|G′}}, and let {{Mvar|H}} be the quotient group {{Math|G′ / G′′}}. Then we say that {{Mvar|H}} is a subquotient of {{Mvar|G}}. In symbols, let {{Math|G′′ ◃ G′ ≤ G}} and {{Math|1=H = G′ / G′′}}; then {{Math|H ⪯ G}}. This relationship has the following properties:

1. Reflexivity: $G\backslash preceq\; G$, i. e. every element is related to itself. Indeed, $G$ is isomorphic to the subquotient $G/\backslash \{1\backslash \}$ of $G$.
2. Antisymmetry: if $G\backslash preceq\; H$ and $H\backslash preceq\; G$ then $G\backslash cong\; H$; that is, no two distinct elements precede each other. Indeed, a comparison of the group orders of $G$ and $H$ then yields $|G|\; =\; |H|$ from which $G\backslash cong\; H$.
3. Transitivity: if $H\text{'}/H$ \preceq H and $H\backslash preceq\; G$ then $H\text{'}/H$ \preceq G.

Let $H\text{'}/H$ be a subquotient of $H$, let $H\; :=\; G\text{'}/G$ be a subquotient of $G$, and let $\backslash varphi\; \backslash colon\; G\text{'}\; \backslash to\; H$ be the canonical homomorphism. Then in the following diagram, all vertical ($\backslash downarrow$) maps $\backslash varphi\; \backslash colon\; X\; \backslash to\; Y,\; \backslash ;\; x\; \backslash mapsto\; x\; \backslash ,\; G\text{'}\text{'}$

are surjective for the respective pairs

The preimages $\backslash varphi^\{-1\}\backslash left(H\text{'}\backslash right)$ and $\backslash varphi^\{-1\}\backslash left(H$\right) are both subgroups of $G\text{'}$ containing $G$ , and it is $\backslash varphi\backslash left(\backslash varphi^\{-1\}\backslash left(H\text{'}\backslash right)\backslash right)\; =\; H\text{'}$ and $\backslash varphi\backslash left(\backslash varphi^\{-1\}\backslash left(H$\right)\right) = H, because every $h\; \backslash in\; H$ has a preimage $g\; \backslash in\; G\text{'}$ with $\backslash varphi(g)\; =\; h\; .$ Moreover, the subgroup $\backslash varphi^\{-1\}\backslash left(H\text{'}\text{'}\backslash right)$ is normal in $\backslash varphi^\{-1\}\backslash left(H\text{'}\backslash right)\; .$

As a consequence, the subquotient $H\text{'}/H$ of $H$ is a subquotient of $G$ of the form $H\text{'}/H$ \cong \varphi^{-1}\left(H'\right)/\varphi^{-1}\left(H''\right) .

In constructive set theory, where the law of excluded middle does not necessarily hold, one can consider the relation subquotient of as replacing the usual order relation(s) on cardinals. When one has the law of the excluded middle, then a subquotient $Y$ of $X$ is either the empty set or there is an onto function $X\backslash to\; Y$. This order relation is traditionally denoted $\backslash leq^\backslash ast\; .$ If additionally the axiom of choice holds, then $Y$ has a one-to-one function to $X$ and this order relation is the usual $\backslash leq$ on corresponding cardinals.

• Homological algebra
• Subcountable

Category:Category theory

Category:Abstract algebra