Finitary alternating group is intermediately monolith in symmetric group
Let be an infinite set, or a finite set of cardinality at least . Then, the Finitary alternating group (?) on is a Monolith (?) (a nontrivial normal subgroup contained in every nontrivial normal subgroup) in the Symmetric group (?) on .
In fact, more is true: it is a monolith in every subgroup of the symmetric group containing it.
- Alternating groups are simple: The alternating group on a finite set of size at least is simple.
- Finitary alternating groups are simple: The finitary alternating group on an infinite set is simple.
- Simple normal implies minimal normal
- Finitary alternating group is centralizer-free in symmetric group
- Self-centralizing and minimal normal implies monolith
- Normality satisfies intermediate subgroup condition
Given: A set of cardinality at least . is the finitary alternating group on . is any subgroup of the symmetric group on , containing .
To prove: is contained in every nontrivial normal subgroup of .
Proof: Note that since is normal in the whole symmetric group on , is also normal in by fact (6). Further, by facts (1) and (2), we see that is a simple group in all cases. Thus, is a simple normal subgroup of . By fact (3), is a minimal normal subgroup of . By fact (4), we conclude that is trivial, so is, in particular, a self-centralizing subgroup of . Fact (5) then tells us that is a monolith in .