Finitary alternating group is intermediately monolith in symmetric group

From Groupprops
Jump to: navigation, search

Statement

Let S be an infinite set, or a finite set of cardinality at least 5. Then, the Finitary alternating group (?) on S is a Monolith (?) (a nontrivial normal subgroup contained in every nontrivial normal subgroup) in the Symmetric group (?) on S.

In fact, more is true: it is a monolith in every subgroup of the symmetric group containing it.

Facts used

  1. Alternating groups are simple: The alternating group on a finite set of size at least 5 is simple.
  2. Finitary alternating groups are simple: The finitary alternating group on an infinite set is simple.
  3. Simple normal implies minimal normal
  4. Finitary alternating group is centralizer-free in symmetric group
  5. Self-centralizing and minimal normal implies monolith
  6. Normality satisfies intermediate subgroup condition

Proof

Given: A set S of cardinality at least 5. H is the finitary alternating group on S. G is any subgroup of the symmetric group on S, containing H.

To prove: H is contained in every nontrivial normal subgroup of G.

Proof: Note that since H is normal in the whole symmetric group on S, H is also normal in G by fact (6). Further, by facts (1) and (2), we see that H is a simple group in all cases. Thus, H is a simple normal subgroup of G. By fact (3), H is a minimal normal subgroup of G. By fact (4), we conclude that C_G(H) is trivial, so H is, in particular, a self-centralizing subgroup of G. Fact (5) then tells us that H is a monolith in G.