Help:Subgroup metaproperty satisfaction lookup

This page gives information on finding statements and proofs of metaproperty satisfactions and dissatisfactions by subgroups. The two reference categories are:

• Category:Subgroup metaproperty satisfactions
• Category:Subgroup metaproperty dissatisfactions

Formulating and looking up a subgroup metaproperty satisfaction/dissatisfaction

The idea behind formulation

The following statements, though they do not mention subgroup properties or metaproperties, are examples of subgroup metaproperty satisfactions:

1. If $H\leq K\leq G$ are groups, and $H$ is a normal subgroup of $G$, then $H$ is a normal subgroup of $K$.
2. If $H\leq K\leq G$ are groups, such that $H$ is a characteristic subgroup of $K$ and $K$ is a characteristic subgroup of $G$, then $H$ is a characteristic subgroup of $G$.
3. If $H$ and $K$ are subgroups of a group $G$ such that $H$ is normal in $G$, then $H\cap K$ is normal in $K$.

Here are some examples of subgroup metaproperty dissatisfactions:

1. It can happen that $H\leq K\leq G$ are groups such that $H$ is normal in $K$ and $K$ is normal in $G$, but $H$ is not normal in $G$.
2. If $H\leq K\leq G$ are groups, such that $H$ is a characteristic subgroup of $G$, it is not necessary that $H$ be a characteristic subgroup of $K$.

There are two main steps to identifying a subgroup metaproperty satisfaction/dissatisfaction:

• First, identify the subgroup property that the statement is about.
• Second, identify the statement that is being made about the subgroup property, and abstract it into a subgroup metaproperty.

For the second step, it is important to know the names (as used in this wiki) for the important subgroup metaproperties. (In most cases, there is no standard terminology, though the names used on the wiki are by and large self-explanatory). Here are some examples:

1. If $H\leq K\leq G$ with $H$ normal in $G$, then $H$ is normal in $K$: The subgroup property here is normal subgroup and the subgroup metaproperty here is the intermediate subgroup condition. A subgroup property $p$ satisfies the intermediate subgroup condition if whenever $H$ satisfies property $p$ in $G$, $H$ also satisfies property $p$ in every intermediate subgroup $K$. The proof of the statement is found at Normality satisfies intermediate subgroup condition.
2. If $H\leq K\leq G$ with $H$ characteristic in $K$, $K$ characteristic in $G$, then $H$ is characteristic in $G$: The subgroup property here is characteristic subgroup and the metaproperty is called being a transitive subgroup property. The proof of the statement is found at characteristicity is transitive.

Hunting for metaproperty satisfactions starting from the property page

If hunting for metaproperty (dis)satisfactions about the property of being a normal subgroup, the first place you should go is the page on normal subgroup. There are now two approaches:

• Go to the metaproperties section. If you know what the metaproperty is called, go right to that subsection; otherwise, read through all the subsections. Each subsection is about a particular metaproperty, and states whether the subgroup property satisfies that metaproperty. In many cases, a link to a separate page with more explanation and proof is provided.
• At the top of the page, in the box stating that the property is a subgroup property, there may be a link under VIEW RELATED to subgroup metaproperty satisfactions and subgroup metaproperty dissatisfactions. These links provide automatically generated lists.

Sources of confusion

Not every statement about a subgroup property qualifies as a metaproperty satisfaction or dissatisfaction. Here are some other kinds of statements:

• Subgroup property implications/non-implications: These assert that one subgroup property implies/does not imply the other. Lists are available at Category:Subgroup property implications and Category:Subgroup property non-implications. Further information: Subgroup property implication lookup
• Subgroup property operator computations: These make statements about more complex relations between multiple subgroup properties. The category Category:Subgroup property operator computations has lists of different kinds of operator computations. For instance, there's Category:Composition computations, which involves statements like: If $H\leq K\leq G$ are groups such that $H$ has property $p$ in $K$ and $K$ has property $q$ in $G$, then $H$ has property $r$ in $G$.

Going through all these lists may be painful. Luckily, there's a one-stop method for viewing all facts related to a subgroup property. This is usually linked to from a box at the top, which describes the type of term. A line in the box says: VIEW: Definitions built on this | Facts about this ...

The list of facts about any term can also be obtained directly by going to:

https://groupprops.subwiki.org/wiki/Special:SearchByProperty/Termname

For instance, all facts related to normal subgroup can be accessed by:

https://groupprops.subwiki.org/wiki/Special:SearchByProperty/Normal_subgroup

More sophisticated searches are possible as well. Further information: Help:Semantic search