The classification all finite groups which possess a single proper non-trivial normal subgroup

We know that

For n≥5, A_n is the only proper nontrivial normal subgroup of S_n.

I am kindly asking to know the possible presented references including the following point, if anybody is aware of them.:

The classification all finite groups G whose possess a single proper non-trivial normal subgroup.

Thanks for your time.

Answer

You will get similar examples by taking a finite, nonabelian simple group S, and extending it by an outer automorphism of prime order to a group G.

Somewhat dually, you can take a quotient P of prime order of the Schur multiplier of S, and extend P by S to a group G.

Another class of soluble examples can be obtained by starting with two distinct primes p, q. Consider the period n of p modulo q. Then in the multiplicative finite field \mathbf{F}_{p^{n}} there is a subgroup Q of order q that acts irreducibly on the additive group P of \mathbf{F}_{p^{n}}. The semidirect product G = PQ will have the property, with P the only nontrivial, proper normal subgroup.

Coming back to insoluble examples, one can take a direct power S^{p} of a nonabelian, finite simple group S, with p prime. If you let a cyclic group C_p of order p permute cyclically the factors in S^{p}, you should get another example, with S^{p} as the distinguished normal subgroup.

Attribution
Source : Link , Question Author : Mikasa , Answer Author : Andreas Caranti

Leave a Comment