Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

Contents 1 Statement 2 Facts used 3 Proof 4 References 4.1 Textbook references

Statement

Suppose G is a primitive group, M is a core-free maximal subgroup of G (or, the stabilizer of a point for a faithful primitive group action of G), and A is an abelian normal subgroup of M that is a contranormal subgroup of G: G is generated by the conjugates of A in it. Then, the commutator subgroup [G,G] is the monolith of G, i.e., every nontrivial normal subgroup of G contains the commutator subgroup of G.

Facts used

1. Second isomorphism theorem
2. Abelianness is quotient-closed

Proof

Given: A group G, a core-free maximal subgroup M of G. An abelian normal subgroup A of M that is contranormal in G. A nontrivial normal subgroup N of G.

To prove: G / N is abelian.

Proof:

1. MN = G: Since M is core-free and N is nontrivial normal, M does not contain N. Since M is maximal, MN = G.
2. AN is normal in G: Clearly, . Also, A is normal in M and N is normal in <mah>G</math>, so . Thus, , so N_G(AN) = G by step (1).
3. AN = G: By assumption, A is contranormal. Thus, the only normal subgroup of G containing A is G. So, by step (2), AN is normal in G.
4. is abelian: This follows from facts (1) and (2).

References

Textbook references

• Book:Cohn, Page 120, ^{More info}